Structure of electronic materials

CHAPTER

Structure of electronic materials

1

CONTENTS 1.1 Atomic Bonding in Metals, Semiconductors, and Dielectrics .................................. 1 1.2 Symmetry of Crystals ......................................................................................... 12 1.3 Basic Structures of Crystals Used in Electronics ................................................. 26 1.4 Lattice Defects in Crystals ................................................................................. 37 1.5 Structure and Symmetry of Quasicrystals and Nanomaterials ............................... 47 1.6 Structures of Composites and Metamaterials ....................................................... 56 1.7 Summary .......................................................................................................... 62 References .............................................................................................................. 68

Before describing the most diverse properties of materials used in electronics, it is necessary to consider the features of their structures, in which these properties are realized. The formation of crystal, amorphous, and other substances from atoms is accompanied by a decrease of the energy in a system as compared to unconnected atoms. The minimum energy in solids corresponds to a regular arrangement of atoms that agrees with the specific distribution of electronic density between them. In accordance with the electronic theory of valence, interatomic bonds are formed due to the redistribution of electrons in their valence orbitals, resulting in a stable electronic configuration of noble gas (octet) due to formation of ions or of shared electron pairs between atoms.

1.1 ATOMIC BONDING IN METALS, SEMICONDUCTORS, AND DIELECTRICS Any connections of atoms, molecules, or ions are conditioned by electrical and magnetic interactions. At longer distances, electrical forces of attraction dominate between particles whereas, at short distances, repulsion of particles increases sharply. The balance between such long-range attraction and short-range repulsion is the cause of the basic properties of substances. The atomic connection is attributable to the restructuring of atomic electronic shells, thus creating chemical bonds. In other words, chemical bonds are the phenomenon of atomic interaction by means of overlap of their electronic clouds, and this is accompanied by a decrease of the total energy of a system. Electronic Materials. https://doi.org/10.1016/B978-0-12-815780-0.00001-3 # 2019 Elsevier Inc. All rights reserved.

1

2

CHAPTER 1 Structure of electronic materials

Chemical bonding is characterized by both energy and length. A measure of bond strength is the energy, expended in case of bond destruction, or the energy gained during compound formation from individual atoms. Consequently, the energy of chemical bonds equals the work that must be expended to separate particles that are constrained, or to alienate them from each other on the infinite distance [1]. During the formation of chemical bonds, exactly those electrons that belong to the valence shells play a major role because their contribution to solid body formation is much greater than that of the inner electrons of atoms. However, division into ionic residues and valence electrons is a matter of convention. For example, in metals it is sufficient to consider that valence electrons are transformed into conduction electrons whereas all other electrons belong to ionic residues. In the atoms of a metal, their outer electronic orbits are filled with a relatively small number of electrons that have low ionization energy. When these atoms come together (i.e., when crystal is formed from atoms), the orbits of valence electrons strongly overlap. As a result, valence electrons in metals become uniformly distributed in a space between cations, and these electrons have a common wave function. Therefore valence electrons in most metals are fully collectivized, and thus such crystals constitute a lattice of positively charged ions crowded by “electronic gas.” Unlike, for example, covalent bonds, the complete delocalization of electrons is a distinctive feature of metallic bonds. It is in this way that the spatial distribution of valence electrons lies at the heart of the classification of solids (dielectrics, semiconductors, and metals). The division of crystals into different classes suggests that solids consist of: •

•

ionic residues, that is, nuclei themselves and those electrons that are so strongly associated with their nuclei that the residues formed cannot significantly change their configuration as compared with the atom; valence electrons, that is, electrons, the distribution of which, in solids, may differ significantly from the configuration existing in isolated atoms.

The spatial distribution of electronic orbitals of certain atoms has a strong influence on the bond strength and their direction. Fig. 1.1 schematically shows how major electronic orbitals for s-, p-, and d-states of electrons in the atoms might look. Only the s-orbital is characterized by spherical symmetry. In contrast, the p-orbital has a very specific form, and this is especially true for the d-orbitals: their forms are considered to contribute to the specific properties of transition metals. Rare earth metals have f-electrons, and they may play a dual role: as residue electrons of “atomic core” and as “free” electrons (because of their complexity, f-orbitals are not shown in Fig. 1.1). Thus during chemical bond formation, valence electrons play a dominant role because, at crystal formation, their contribution is much greater than that of electrons, which form atomic internal orbitals in the residues. A classification of the possible bonds of particles in crystals is shown in Fig. 1.2. This division is rather conditional, because it corresponds to simplified models. In many cases, the actual bonding is more complicated and often presents as an intermediate case between simple models.

1.1 Atomic bonding in metals, semiconductors, and dielectrics

z

1s

y x 2px

z 2py

z

2pz

z

3dz2

y

y

y x

x

x

z

3dx2–y2

z

y

y 3dyz

x

x z

3dxz

z

y x

3dxy

y x

z

y x

FIG. 1.1 Forms of s-, p-, and d- orbitals: angular dependence of square wave functions.

FIG. 1.2 Various models of atomic bonds in crystals [2].

Molecular and metallic bonds are shown at the opposite sides of the scheme, because they are absolute opposites. In molecular crystals, electrons usually are completely locked in their molecules (or atoms; Fig. 1.3A). The nuclei are surrounded by spaces (shown as black balls), where the density of the electronic cloud reaches significant values. The simplest examples of molecular bonds are atomic crystals of solid inert gases: neon, argon, krypton, and xenon. These have completely filled electronic shells, and such a stable electronic configuration undergoes only minor changes during the formation of solids. Therefore, the inert gas crystal is an example of a rigid body with strong electron bonding exclusively inside atoms, whereas the electron density between atoms is rather small, because all electrons are well localized near their own nuclei.

3

4

CHAPTER 1 Structure of electronic materials

FIG. 1.3 Two-dimensional image of electrical charge distribution: (A) molecular crystal, in which quadrupole electronic fluctuation (+ 2 … 2 +) results in the attraction of atoms, whereas partial overlapping of electronic shells leads to repulsion ( …!), thereby balancing this attraction; (B) metal crystal, black circles represent positively charged atomic residues, immersed in electronic gas.

The metal bond. As already noted, in metal atoms their outer electronic orbitals contain a rather small amount of electrons that have low ionization energy. When such atoms come closer (i.e., when a metal crystal or alloy is formed), orbitals of valence electrons largely overlap each other. As a result, these electrons become distributed almost uniformly in the space between ions (Fig. 1.3B). Indeed, X-ray studies have practically indicated a uniform electronic density in the lattice of metals. Therefore valence electrons in metals are a joint collective in the crystal as a whole, and metal represents the lattice of positively charged ions wherein the “electronic gas” exists. This is a reason for the delocalization of metal bonds; moreover, metal bonds are unsaturated and nondirectional. Metals are, among crystals, characterized by the highest coordination number (CN) of ions (usually in metals, this number is 12; it is the number of the nearest neighbors to a given particle). For comparison, it should be noted that in ionic crystals this number is often 6 or 8. Similarly, the CN in the covalent crystals is even smaller—it is 4 for semiconductors with a diamond structure. The bonding in dielectrics and semiconductors differs significantly from metal bonds. Fig. 1.4 schematically shows the energy dependence on the distance between atoms for bonds in basic types of dielectrics and semiconductors (metal bonds are not shown). Between particles (atoms, molecules, or ions) when creating a semiconductor or dielectric material, at relatively large distances, the forces of attraction dominate: the corresponding energy is negative and characterized by the curve 1. At short distances, the force of repulsion becomes much more powerful; its energy is positive

1.1 Atomic bonding in metals, semiconductors, and dielectrics

FIG. 1.4 Dependence of attraction energy (1), repulsion energy (2), and total energy (3) on the distance between particles r : (A) ionic bond, (B) covalent bond; (C) molecular (quadrupole) bond; and (D) hydrogen bond.

and characterized in Fig. 1.4 by curve 2. The total potential energy of interaction between particles is shown by curve 3 that the minimum energy corresponds to a stable distance between the interacting particles (this is parameter of lattice). The strong repulsion between approaching atoms or ions can be modeled by drastic energy dependence: Urep  r8 … r12; this dependence characterizes the mutual impenetrability of electronic orbitals: electronic shells of neighbor atoms or ions can penetrate each other only very slightly. This is the reason that atoms, ions, or molecules can be presented by the “hard spheres” of certain radii, the size of which remains practically unchanged [3]. The attraction forces that tie atoms, ions, and molecules together in solids are of an electrical nature. It should be noted that crystals are classified just by the nature of attraction forces. As shown in Fig. 1.4, the main types of chemical bonds in dielectrics and semiconductors are the covalent, ionic, molecular, and hydrogen bonds. The metal bond (not shown in Fig. 1.4) can be considered a limiting case of the covalent bond. The ionic bond. Ionic crystals (such as sodium chloride, Na+ Cl) are chemical compounds formed from metal and nonmetallic elements. The energy of ionic attraction varies with distance rather slowly; therefore ionic bonds are the most long

5

6

CHAPTER 1 Structure of electronic materials

FIG. 1.5 Two-dimensional image of electronic charge distribution in: (A) ionic crystal, where ion attraction is balanced by partial overlapping of electronic shells; (B) covalent crystal, black diffused circles represent atomic residues surrounded by regions, where electronic density reaches significant values.

ranging in comparison with others. Similar to atomic or molecular crystals (shown in Fig. 1.3A), ionic crystals can be characterized by such a distribution of electronic charge that is almost completely localized near ions. In the simplest model of the ionic crystal (Fig. 1.5A), ions are “nearly impenetrable charged balls.” This approximation is rather suitable for ions that have completely filled electronic shells. Typically, cations and anions acquire electronic configuration of the inert gas, and therefore the charge distribution in them has an almost spherical symmetry. Ions with opposite charges attract each other due to long-range Coulomb forces; therefore the energy of their attraction varies with distance very slowly: Uatt  r1 (Fig. 1.4A). At the same time, the repulsive energy of ions is inversely proportional to the interatomic distance: Urep  r8 … r12 (depending on the properties of the given crystal). Therefore the ionic crystal can be considered a molecular crystal in which the lattice is built, not from atoms, but from the ions (e.g., ions Na+ and Cl in the rock salt). Thus charge distribution in the ion, located in a solid body, is only slightly different as though it were an isolated ion. It is important that particles, which form ionic crystals are not neutral atoms: between ions, large electrostatic forces exist that play a major role and determine the main properties of ionic crystals (that differ significantly from the properties of molecular crystals). Thus in the simplest model of an ionic crystal, all ions are presented as “interacting nearly impenetrable charged spheres”; this approximation is sufficient for ions with entirely filled electronic shells. Whereas in the atomic (or molecular) crystal, all electrons remain locked in their native atoms, in the ionic crystal, valence electrons are moved from cations to anions. Therefore the ionic bond occurs between particles of two types, one of which easily loses electrons, forming positively charged ions (cations) and other atoms that

1.1 Atomic bonding in metals, semiconductors, and dielectrics

readily get electrons then form, respectively, negatively charged ions (anions). Most of the electropositive cations belong to groups I and II of the periodic table, whereas most anions belong to groups VI and VII. As a rule, ions in crystals are packed tightly, as each of them is surrounded by the largest number of oppositely charged ions. Stabilization of the ionic solid structure takes place at CNs 6, 8, and, occasionally, even 12. It should be noted that ionic radii vary noticeably with the value of the CN. Ionic bonds, unlike metal bonds, are saturated, but, as in metals, they are not directed [4]. The covalent bond in crystals is typical for semiconductors. The dependence of binding energy on interatomic distance is shown in Fig. 1.5B; attractive forces in case of covalent bonds are not so long ranging as in the case of the ionic bond: the attraction energy changes with distance as r2 … r4. In principle, the nature of the covalent bond is very close to that of the metal bond; however, in covalent crystals, valence electrons are shared only between the nearest neighboring atoms whereas, in metals, valence electrons are shared within the crystal lattice. Usually, a covalent bond (i.e., homeopolar bond) is formed with a pair of valence electrons that have opposite spin directions. During covalent chemical bond formation, the reduction of total energy is achieved by the quantum effect of exchange interaction. The simplest example of a covalent bond is the hydrogen molecule H2, wherein both electrons belong simultaneously to both atoms. The diamond might be a classic example of a covalent crystal (Fig. 1.5B), where carbon atoms are located in a rather roomy configuration: their CN is only 4. Therefore, diamond (as with semiconductors of similar structure—germanium and silicon) is characterized by a comparatively high-density electronic cloud in the atomic interstitials: electrons are concentrated mainly near the lines connecting each carbon atom with its four nearest neighbors. Although diamond is dielectric, the high charge density in areas between atoms is a characteristic feature of semiconductors. Covalent bonds, unlike metal bonds, are strongly directed; moreover, they are saturated. The saturation of a covalent bond is the ability of atoms to form a limited number of covalent bonds. The number of bonds formed by the atom is determined by its outer electronic orbital. The directivity of covalent bonds is caused by their peculiar electronic structure and geometrical shape of electronic orbitals (the angles between two bonds are the valence angles). Sometimes, covalent bonding might have pronounced polarity and increased polarizability that determines many chemical and physical properties of correspondent compounds. The polarity of a covalent bond is due to the uneven distribution of electronic density accruable to a difference in the electronegativity of atoms (therefore covalent bonds are divided into nonpolar and polar bonds). The polarizability of bonds can be expressed by the nonsymmetric spontaneous displacement of binding electrons [5]. The following polar connections are distinguished [4]: •

The nonpolar (simple) covalent bond arises from the fact that each atom provides one of its unpaired electrons, but the formal charge of atoms remains unchanged, because atoms that form the bond equally have a socialized electron pair.

7

8

CHAPTER 1 Structure of electronic materials

•

•

•

The polar covalent bond, wherein atoms are different, the degree of overlap of the socialized pair of electrons is determined by the difference in the electronegativity of atoms. The atom with a greater electronegativity more strongly attracts electrons; therefore its real charge becomes more negative. With the electronegative charge an atom acquires, there is additional positive charge of the same magnitude. The donor-acceptor bond arises when both connecting electrons are provided by one of the atoms (called the donor) whereas the second atom, involved in the formation of a bond, is the acceptor. When creating this pair, formal charge of the donor is increased by one and the formal charge of the acceptor is reduced by one. The electron pair of one atom (donor) goes into a common use, whereas another atom (acceptor) provides its free orbital, because the donor atoms usually serve atoms that have more than four valence electrons. The σ-bond and π-bond are approximate descriptions of some types of covalent bonds in different compounds. Therefore the σ-bond is characterized by the maximum electronic cloud density along the axis joining the nuclei of atoms. The formation of the π-bond is characterized by the lateral overlap of electronic clouds “above” and “below” the plane of the σ-bond.

Unlike metallic coupling, the emergence of a covalent bond is accompanied by such redistribution of electronic density that its maximum localizes between the interacting atoms. As in metals, in case of a covalent bond, the collectivization of the outer valence electrons is seen, but the nature of electronic allocation is different from that in metals. In the ground state of covalent crystals, that is, at T ¼ 0 K, there are no partially filled electronic energy bands. In other words, the covalent crystal cannot be described by uniform distribution of electronic density between atoms, as is typical for simple metals. Conversely, in covalent crystals, the electronic density is increased along the “best destinations,” leading to chemical bonds. The stronger the covalent bond, the greater the overlap of electronic clouds of interacting atoms. If this bond is formed between similar atoms, the covalent bond is the homeopolar and, when atoms are different, it is the heteropolar. In cases where two interacting atoms share one electron pair, a single connection is formed; when there are two electron pairs, the double bond is created, and when there are three electron pairs, a triple bond is created. The distance between bound nuclei is defined as the length of the covalent bond. Bond length decreases when the order of the bond increases. For example, the length of a “carbon-to-carbon” bond depends on multiplicity: for a CdC bond, its length is 1.54  101 nm; in case of a C]C bond, the length is 1.34  101 nm, whereas for C^C, it is only 1.20  101 nm [3]. With an increase of the bond order, its energy increases. The directivity of covalent bonds characterizes the features of electronic density distribution in atoms. For instance, in germanium and silicon crystals (that have a diamond structure), each atom is located in the center of a tetrahedron, formed by

1.1 Atomic bonding in metals, semiconductors, and dielectrics

FIG. 1.6 Structure of main semiconductors: diamond (A), sphalerite (B), and wurtzite (C) [6].

four atoms, their closest neighbors (Fig. 1.6A). In this case, tetrahedral bonds are formed when each atom has only four nearest neighboring atoms. Most covalent bonds are created by two valence (hybridized) electrons—one from each interacting atom. In case of such a connection, the electrons are localized in the space between the two atoms; thus the spins of these electrons are antiparallel. As shown in Fig. 1.5B, the plane scheme can give only an approximate representation of the actual location of atoms. In fact, the relative position of these atoms in real crystals can be quite complex, as shown in Fig. 1.6B and C. The structure of the mineral sphalerite (zinc sulfide, ZnS) is typical of AIIIBV semiconductors, such as gallium arsenide. The wurtzite structure (calcium selenide, CaSe) is typical of AIIBVI semiconductors. Simplified schemes of electronic density distribution in covalent and ionic crystals are shown in Fig. 1.7A and B. However, sphalerite and wurtzite belong to polar crystals that have a hybrid bonding. The hybrid ionic-covalent bond. As with the model of a “purely covalent” structure, a model of a “purely ionic” crystal is idealized. In real crystals (especially, in some AIIIBV and AIIBIV types of semiconductors and in active dielectrics), the intermediate case between ionic and covalent bonds exists. In the covalent silicon crystal (Fig. 1.7A), electrons are equally distributed around atoms; therefore, the electronic density between atoms is rather large. In an ionic crystal, the attraction of cation and anion is compensated by the repulsion of partially overlapping electronic shells. The concept of intermediate type bonds agrees with the theory of ion deformation by their polarization. This may occur, for example, by the distortion of an anion’s electronic orbital, mainly by the different electronegativities of adjacent ions. Therefore, the electronic density between ionic residues increases, that is, the mixed covalent-ionic bond with a greater degree of charge separation becomes the polar bond. The exact presence of such bonds determines the noncentrosymmetric structure of some crystals. The hybrid ionic-covalent bonding is the main cause of pyroelectric, ferroelectric, and piezoelectric properties. Most such active (functional) dielectrics belong to crystals or to other ordered polar systems (liquid crystals, electrets, polar polymers, etc.). Thus the physical hypothesis, relevant to the nature of the

9

10

CHAPTER 1 Structure of electronic materials

FIG. 1.7 Simplified scheme of transition from covalent and ionic bonds to mixed polar bond: (A) in covalent bond, electronic density (ρ) distribution is quite symmetric (arrows symbolize opposite orientation of spins in connecting electron pair); (B) in ionic bond, cation and anion are attracted (big arrows), whereas a small overlap of electronic shells ensures repulsion (small arrows); electronic density distribution is almost symmetric, (C) asymmetric mixed bond that leads to polar properties of crystals, wherein both attraction and repulsion are seen: a covalent bond is formed by electron pair with opposite spins; electronic density distribution is asymmetric, and can be characterized by displacement δ.

internal polarity (which is not caused by an external electrical field), deserves particular attention. This hidden (or latent) polarity manifests itself in polar crystals as the ability to provide electrical (vectorial) response to any nonelectrical scalar, vector, or more complicated tensor impacts [5]. The tendency of polar crystals to generate an electrical response on nonelectrical impact leads to their generation of electrical potential under uniform heating of crystals (pyroelectricity) or under uniform deformation (piezoelectricity). These are mostly crystals with hybrid ionic-covalent bonding. Exactly this peculiarity causes a reduction in crystal symmetry; therefore polar crystals always belong to noncentrosymmetric classes of symmetry.1 It is obvious that the primary cause of the peculiarities of polar crystals is the asymmetry of electronic density distribution along atomic bonds. The fundamental reason for this asymmetry is a distinction in the electronegativity of atoms (a physical property that describes the tendency of an atom to attract electrons). Electronegativity depends on atomic number, as well as on the size and structure of outward (valence) electronic orbitals [1]. The higher the atomic electronegativity, the stronger the aptitude of atoms to attract electrons toward themselves. 1 Comments. In contrast, crystals with exclusively ionic bonds as well as crystals with exclusively covalent bonds are nonpolar. Usually, they belong to the centrosymmetric classes of crystals: in typically ionic crystals, a central symmetry exists, and there are no special orientations in atomic connections. In the same way, simple covalent crystals also belong in centrosymmetric structures: each atom provides for bond one unpaired electron; thus four socialized electron pairs are located symmetrically.

1.1 Atomic bonding in metals, semiconductors, and dielectrics

The difference of atoms by electronegativity might be very substantial. Therefore atoms with higher electronegativity strongly attract conjunctive electrons, and its true charge becomes more negative. Conversely, the atom with lower electronegativity acquires an increased positive charge. Together, these atoms create a polar connection and, correspondingly, the noncentrosymmetric structure. Simultaneously, such connections do not lead to the appearance of internal fields, but can provide a peculiar response to external impact that is quite different in various noncentrosymmetric crystals. For example, in case of directional mechanical influence onto a polar crystal, an electrical response arises (piezoelectric effect). The point here is that the elastic displacement of atoms compresses (or stretches) their asymmetric connections, and thereby induces electrical charges on the crystal surface (piezoelectric polarization). In contrast, if atomic connections in crystal are centrosymmetric, no electrical response is possible to any uniform mechanical impact (however, the inhomogeneous thermal or mechanical impact makes atomic bonds asymmetric, which results in the appearance of an electrical response in any crystal). The fact is that, in many crystals (e.g., in various semiconductor compounds), the type of bonding has an intermediate character between covalent and ionic. It is noteworthy that under conditions of very high pressure, any material with ionic or covalent bonding would acquire the property of a metal bond, and the material would turn into a metal. Thus very high pressure leads to a forced convergence of atoms with great overlap of their outer electron shells. (It should be noted that, in some rare cases, even at normal pressure, a phase transition of “dielectric-metal” is possible; this transition might be stimulated by temperature change or by an external electrical or magnetic field) [5]. The energy of ionic, covalent, and metallic chemical bonding is characterized by similar orders of magnitude. In this respect, they are much inferior to molecular bonds. Molecular bonds (van der Waals bonds) always exist, but only when much stronger valence bonds are absent do these molecular bonds become the main type of chemical connection, primarily, in molecular crystals. Forces of attraction in this case are relatively small, being short range: the energy of intermolecular attraction varies with distance as Uatt  r4 … r6 (Fig. 1.4C). It is evident that this kind of attraction is weak in comparison with ionic and covalent forces; therefore van der Waals bonds are additive and nonsaturated. In case of nonpolar molecules, the forces of attraction are due to the accidental deformations of electronic shells. Quantum fluctuations of electronic density in molecules always exist; thereby, virtual electrical dipoles (or quadrupoles) lead to molecular attraction (in Fig. 1.4C, van der Waals bonding is shown schematically and only as dipole-to-dipole interaction). The electronic polarizability of orbitals determines optical dispersion in materials; therefore forces of attraction of this type, sometimes, are called dispersion forces. In case of polar molecules, the orientation interaction also contributes to the usual molecular interaction. The influence of a molecule’s intrinsic (permanent) dipole onto the induced dipole of another molecule is the inductive interaction.

11

12

CHAPTER 1 Structure of electronic materials

In general, in case of van der Waals bonding, the main contribution is provided by the dispersive forces; however, when molecules have large dipole movements, the contribution of the orientation effect might be significant. As a rule, the inductive interaction is negligible [4]. The hydrogen bond appears between hydrogen atoms and the electronegative atoms P, O, N, Cl, and S belonging to other molecule. The nature of this bond lies in the redistribution of electronic density between atoms, conditioned by the hydrogen ion H+ (proton; Fig. 1.4D). Crystals with hydrogen bonds (dielectrics and semiconductors) show properties similar to molecular crystals, but there is a reason to allocate them to a special class. Hydrogen is unique in the following respects: • •

•

the residue of hydrogen ion is a “bare” proton measuring approximately 1013 cm (i.e., 105 times smaller than any other ion); hydrogen needs only one electron to constitute a stable helium type; the electronic shell (unique among other stable configurations having only two electrons in the outer shell); the ionization potential (energy required to remove an electron from an atom) in hydrogen is high: 13.6 eV (in alkali-halide metals, it is 4 eV).

Because of these properties, during crystal structure formation, the effect of hydrogen may differ significantly from the influence of other elements. Due to the high ionization potential of the hydrogen atom, it is difficult to completely remove its lone electron. Therefore the formation of ionic crystals with hydrogen occurs differently than, for example, in the case of alkali-halide metal crystals [2]. The hydrogen atom may not behave in a crystal as a typical covalent atom: when the H atom loses its electron, it can create only a single covalent bond, shared with another atom. Because the size of the proton is approximately 1013 cm, it is localized in the surface of large negative ions; therefore such a structure arises, which cannot be formed with any other positive ions. The energy of the hydrogen bond is less by order of magnitude than the energy of the covalent bond, but it is greater than the energy of van der Waals interactions in the order of magnitude. Although hydrogen bonds are not very strong, they play an important role in the properties of correspondent crystals. The hydrogen bond is directional; molecules that form the hydrogen bond tend to have a dipole moment that indicates the polar nature of this bonding. In some crystals, the hydrogen bond leads to their piezoelectric, pyroelectric, and ferroelectric properties. Furthermore, it should be noted that molecular and hydrogen bonds are very important in various structures of liquid crystals.

1.2 SYMMETRY OF CRYSTALS In many solids, structural symmetry plays a crucial role for the explanation of properties. Different materials are most frequently used in electronic special effects in crystals, polycrystalline materials, and polymeric films due to the peculiarities of their macro- and microsymmetries.

1.2 Symmetry of crystals

A crystal is a body that, due to its intrinsic properties, is limited by flat surfaces— crystal faces. A more complete definition of crystal should characterize such peculiar intrinsic properties that distinguish crystallized substance from amorphous materials and can explain the multifaceted shape of a crystal. The relationship between the outward geometry and the internal structure of crystal and its physical properties is settled by crystallography. It studies the physical properties of crystals through a specific method—the symmetry that connects physical properties of crystals with their structure. The physics of crystals formulates certain principles that establish a community of crystal symmetry and their physical phenomena; these major principles were advanced by Neumann and Curie [7]. The manifestation of symmetry in geometric forms is the ability of the shape to regularly repeat its parts. In other words, the reason for a geometrically correct external crystal shape is the regularity of the internal structure that lies in the spatial lattice of a crystal. This spatial lattice is the abstraction that allows the description of proper and regular alternation of atoms, molecules, or ions, and results in the macroscopic shape of the crystal. This lattice is infinite, and it is constructed by the translation of the unit cell of the crystal along crystallographic coordinates by endless repetition in a space with identical structural units. As a simple example, Fig. 1.8 shows various two-dimensional (2D) translations of unit cells on the surface. All imaging pairs of vectors ai and bi are lattice translation vectors, but they are not primitive vectors. The metric of the unit cell of crystal is determined by the ideal distance between the nearest atoms or ions of a similar kind. In most simple crystals, for example, in the majority of metals the structural unit consists of only a single atom. In dielectric crystals, the unit cell may comprise a plurality of atoms, ions, or molecules. The crystal lattice may be built as a result of unit cell translational transformation, in other words, by various point symmetry operations [6]. The symmetry elements and operations. To describe the symmetry of a crystal’s physical properties as well as to determine the symmetry of geometric forms, a quite ordinary idea is to consider only single space elements (the unit cell of crystal). In the theory of symmetry, the object of study is the figure, that is, a certain set of spatial points. The imaginary geometrical object, over which symmetry operations are performed, is the symmetry element of the finite figure. As symmetry elements, the planes, axis, and center of symmetry (center of inversion) may be used.

FIG. 1.8 Unit cells in a two-dimensional lattice.

13

14

CHAPTER 1 Structure of electronic materials

The symmetry operation involves combining the point (or part of a figure) with another point (or part of the figure). Both combined parts of figure are symmetric. The point symmetry operation should be left in place at least on one point of the figure. It is the intersection point of all elements of symmetry. If symmetry operations are applied to three-dimensional (3D) figures, the twists and turns as well as the inverted turns and the reflection in a plane of symmetry are selected. The symmetry elements are distinguished as first and second types. The former includes the symmetry plane, rotary axis of symmetry, and the center of inversion (center of symmetry). Complicated symmetry elements, such as inversion axis and rotary-reflection axis, belong to the second type of symmetry elements. The symmetry plane is a mirror-reflecting plane that provides a combination of symmetrically equal points; when recording symmetry elements of a particular class of a crystal, the plane of symmetry can be referred to as P. For example, the mirror plane, being a plane in the cube diagonal, divides the cube into two equal mirrormating parts. In the international system, the mirror plane is represented by the letter m. It perpendicularly bisects all segments, connecting balanced (symmetrically equal) points. The rotation symmetry axis of n-order is denoted as Ln. When a figure turns by a specific angle of α ¼ 360ο/n (called elementary angle), a superposition of symmetric points (equal compatible) can be realized. The rotary axes are denoted as 1, 2, 3, 4, 5, 6, 7, …, ∞, where the numbers indicate the order of axis. For example, Fig. 1.9 shows a set of elements of the symmetry of a cube, which has a center of symmetry (in the cube geometric center), three axes 4 (fourth order), four axes 3 (third order), six axes 2 (second order), three planes of symmetry parallel to the faces of a cube, and six diagonal planes of symmetry. Due to the large amount of symmetry elements, the crystals of cubic symmetry are called highly symmetric. Other classes of crystals have much smaller number of symmetry elements. If an arbitrary figure, and not a crystal, is considered, there could be any order of the rotary axis. For example, the sphere has an infinite number of rotational axes, including axes of infinite order. The cylinder has a single axis of infinite order and an infinite number of axes of order 2 (Fig. 1.10). In certain cases, combining of a figure with its initial position must be made not only by elementary rotation angle, but also by the auxiliary reflection plane, perpendicular to the axis about which the figure rotates. The complex axis (or axis of complex symmetry) is the mirror-rotary axis Lni. Operations that function by mirror-rotary axes can be implemented with the help of an inversion axis (denoted also as Lni). The order of the rotary axis of the crystal and mirror rotary axes is strictly limited. These axes can be only of the first, second, third, fourth, and sixth orders. If there are several symmetric axes, the axis with an order higher than 2 is the principal. Both ends of a rotary symmetry axis might be different, in this case, it is the polar axis; for instance, in Fig. 1.11A, the polar axis 4 extends through a tetragonal pyramid. Polar axes are typical of a certain type of crystals (non centrosymmetric classes). As shown in Fig. 1.11, the b plane of symmetry m is perpendicular to the axis 4;

1.2 Symmetry of crystals

FIG. 1.9 Elements of symmetry of cube: axes of symmetry are numbered whereas planes of symmetry are denoted by the letter m.

FIG. 1.10 Geometric figures representing the limiting symmetry group [7].

15

16

CHAPTER 1 Structure of electronic materials

FIG. 1.11 Polar and bipolar rotary axes of 4th order: (A) tetragonal pyramid; (B) tetragonal bipyramid; and (C) tetragonal prism.

in this case, the symmetry of a figure is referred to as 4/m. If the axis lies in the plane of symmetry, delimiters are not needed: 4m. To refer to the symmetry of various crystals, it is possible to use designations: m, 2m, 3m, 4m, and 6m. Markings of the first-order axis of symmetry are not used: that is, “1” near the sign “m” is not needed as an axis of symmetry of first order is always present (when one turns the figure on 360o, any figure will coincide). Besides the usual symmetry axes, the inversion axes exist. Such an axis of order n (axis Lni) combines the joint action of a rotary axis and inversion center. The center of symmetry (the inversion center) is a singular point inside a shape (or inside the unit cell) that is characterized by the fact that any straight line drawn through the center of symmetry (denoted by symbol C) meets the same (respectively) point figures on the opposite side of the center at equal distances. A symmetric transformation in the center of symmetry is the mirror image point (Fig. 1.11B). At this point, as in a photographic lens, the image is inverted. Sometimes, two symmetrically equal figures cannot be superposed other than by reflection. For example, in Fig. 1.12, two molecules of an organic compound are displayed (these do not have rotational axes of symmetry). The figures that can be superposed with each other only by the mirroring are the enantiomorphous ones. The phenomenon of enantiomorphism in crystals is expressed by the formation of left and right forms (e.g., in quartz crystal), which reflects “enantiomorphism” in its physical properties. For example, in the left form of crystals, the rotation of the polarized light plane is clockwise, whereas in right form, it is counterclockwise. This phenomenon is important from the viewpoint of the practical use of such crystals. The concept of the “symmetry element” is broader than the concept of “symmetry operation.” The symmetry element includes all degrees of operation. For example, the axis of symmetry 4 (otherwise denoted L4) implies a set of operations, including

1.2 Symmetry of crystals

FIG. 1.12 Enantiomorphism in certain molecules [6].

40 ¼ 1, 41 ¼ 4, 42 ¼ 2, and 43 ¼ 41. The first operation is the operation of identification, the second is the turn on 90°, the third is the turn on 180° (turns on 180° in opposite directions are equivalent), and operation 43 is the turn on 270° in a certain direction, equal to the rotation in the opposite direction 90° (41). The classes of symmetry are characterized by a set of crystal symmetry elements that describes a possible symmetric transformation. For each crystal, the unit cell can be chosen from which the whole crystal lattice can be built via translations (translations are such displacements that can multiply the unit cell to create a crystal). As a simple example, Fig. 1.13 illustrates some 2D cells (2D lattice). On a plane, each unit cell is defined by two axes (the basis vectors), from which the basic elementary parallelogram can be constructed. Such parallelograms must fill the entire plane of the 2D crystal with no gaps. It is important to note that, in the 2D crystal, only five different types of lattices are possible with a different set of symmetry elements (such elementary lattices are the Bravais lattices). In the 3D space, the unit cell of a crystal lattice is the parallelepiped, built on three basic vectors (Fig. 1.14). The points of intersection of base vectors, composing

FIG. 1.13 Basic two-dimensional lattices: (A) square with j aj ¼ j bj, φ ¼ 90°; (B) hexagonal with j aj ¼ j b j, φ ¼ 120°; (C) rectangular, j a j 6¼ j bj, φ ¼ 90°; and (D) centric rectangular (axes are shown as for primitive, so for rectangular unit cells j aj 6¼ j b j and φ ¼ 90°).

17

18

CHAPTER 1 Structure of electronic materials

FIG. 1.14 Choice of unit cells for different classes of crystals: (A, B, and C) elementary translations on X, Y, and Z directions (called crystallographic coordinate system); α—angle opposite to X-axis; β—angle opposite Y axis; and γ—angle opposite Z-axis.

the spatial lattice, are the junctions. A junction may be located, as in material particles, in the center of gravity of the particle (or group of particles). As in the 2D (plane) lattice grid, the volumetric 3D primitive unit cell of a crystal does not depend on its shape and it has a constant size for a given lattice. The spatial crystal lattice is based on parallel transport of the unit cells that are touching each other by whole faces, filling the entire space without gaps. Thus the choice of elementary translations is not unique; therefore the shortest of these are usually selected that correspond to basis vectors a, b, and c of a lattice. This choice is always carried out by such a way when unit cell would have the maximal number of symmetric elements, and thus can represent the point group of symmetry of the entire lattice. The symmetry of the crystal structure limits the choice of unit cells that can describe it. The choice of the base, and therefore the lattice itself, must comply with the symmetry of crystal structure. All variety of crystals in 3D structures can be described using only 14 types of lattices (Bravais lattices). They differ by the choice of unit cells and are classified by crystal syngony. Therefore three directions, outgoing from a single point of the selected parallelepiped, should be taken as the coordinate axes of a crystal, and thus define the crystallographic axes X, Y, and Z (Fig. 1.14). The rest of the unit cell parameters are the angles between axes: α—between axes Y and Z; β—between axes Z and X; and γ—between axes X and Y. Primitive Bravais cells are the main cells that allow crystal classification by the crystallographic syngonies. Any crystalline structure can be presented with one of 14 Bravais cells listed in Table 1.1. Any linear periodic structure can be obtained by elementary translation. To choose a cell, three guided conditions should be used: • • •

the symmetry of the unit cell must correspond to the highest symmetry of the crystal; the unit cell should have largest possible number of identical angles, or corners and edges; and the unit cell should have the minimal volume.

1.2 Symmetry of crystals

Table 1.1 Fourteen Bravais Cells Lattice type Crystal System (Lattice Basis)

Primitive

BaseCentered

BodyCentered

FaceCentered

Triclinic a 6¼ b 6¼ c; α 6¼ β 6¼ γ 6¼ 90°

Monoclinic a 6¼ b 6¼ c; α ¼ γ ¼ 90°; β 6¼ 90°

Rhombic (orthorhombic) a 6¼ b 6¼ c; α ¼ β ¼ γ ¼ 90°

Trigonal (rhombohedral) a ¼ b ¼ c; α ¼ β ¼ γ 6¼ 90°

Tetragonal a ¼ b 6¼ c; α ¼ β ¼ γ ¼ 90°

Hexagonal a ¼ b 6¼ c; α ¼ 120°; β ¼ γ ¼ 90°

Cubic a ¼ b ¼ c; α ¼ β ¼ γ ¼ 90°

Next, different types of Bravais lattices are usually distinguished as primitive, volume-centered, border-centered, base-centered, and rhombohedral. Division along the crystallographic syngonies determines a choice of the coordinate system and the triples of basis vectors a1, a2, and a3, or, in other words, the metric (γ, β, and α and a, b, or c).

19

20

CHAPTER 1 Structure of electronic materials

Table 1.2 Distribution of Syngonies of Crystallographic Point Groups 1 2 3 4 5 6 7

Syngonies

Classes of Symmetry

Triclinic Monoclinic Orthorhombic Trigonal Tetragonal Hexagonal Cubic

1,1 m, 2/m mm2, 222, mmm 3,3, 3m, 32,3m 4, 4/m, 4mm, 422, 4/mmm,4,42m 6, 6/m, 6mm, 622, 6/mmm,6,62m 23, m3,43m, 432, m3m

According to the syngonies, there are seven types of crystal structures: triclinic, monoclinic, hexagonal, rhombohedral (trigonal), orthorhombic, tetragonal, and cubic. There are 32 classes of symmetry possible in structures with 3D point symmetry, and their distribution in crystal syngonies are shown in Table 1.2 [7]. The items used in this set of symmetry elements are not a random choice but are strictly legitimate mathematical groups. The aggregates of these symmetry elements are introduced according to certain rules. Symbolic images of 10 polar crystal classes are shown further in this work; it is only within these classes that pyroelectric and ferroelectric properties can possibly exist. The principal peculiarity of the lowest category of symmetry is the lack of such symmetry axes that they are greater than the second order. The lowest categories of symmetry are classified as triclinic, monoclinic, and orthorhombic syngonies. The triclinic lattice is the only one that has no elements of rotational symmetry or mirror planes. The lowest-category structures are rarely encountered in nature, and they have lattices based on a cell with three unequal edges and three unequal angles. In crystals belonging to the middle category of symmetry, a symmetry axis obviously exists, which is beyond the second order (major axis). In addition to the major axis, the structure might have a plane m and the center of symmetry C. The middle categories of symmetry are the trigonal, tetragonal, and hexagonal crystal systems. The crystals of highest category of symmetry are crystals of cubic symmetry (cubic system); their main distinguishing feature is the presence of four threefold axes (4L3). Among the many thousands of natural and artificially synthesized crystal structures studied to date, more than half account for the highest category of symmetries. Almost all metals and their alloys crystallize in a cubic system with class m3m or in a hexagonal class 6/mmm. The semiconductors, such as germanium and silicon, are also classified as m3m class; however, the vast majority of semiconductors, including gallium arsenide (GaAs), is related to the point group of cubic symmetry 43m (sphalerite-type structure), as well as to the point group of the hexagonal system 6m2 (wurtzite structure). Almost no substances crystallize in the groups 4, 3, 6, and 432.

1.2 Symmetry of crystals

Connection between symmetry and physical properties. The characteristic features of crystals are their symmetry and anisotropy. The properties of some anisotropic crystal, unlike isotropic crystal, exhibit very high sensitivity to external influences. Influences such as heat, mechanical stress, and electrical and magnetic fields (or adding foreign atoms into crystal) can change conditions of the dynamic equilibrium of the crystal’s constituent particles, that is, they can change the symmetry of the crystal and, thus, change its properties. The ability to manage properties using the abovementioned external influences allows the creation of crystal-based converters of various types of energy. The connection between the physical properties of crystals and their symmetry was formulated by Von Neumann: The symmetry of physical properties of a crystal is not lower than the symmetry of its structure. This means that the structure of crystal, in any case, comprises all elements of the properties’ symmetry (but may also have other symmetric elements). The application of Neumann principle in specific experimental situations was detailed by Pierre Curie. In accordance with the Curie principle, a crystal, being under the impact of external influence, demonstrates such symmetry elements as are common to a crystal in the absence of impact, and the symmetry of impact itself in the absence of a crystal. Thus in the system “crystal-impact,” only the common element of symmetry remains. A geometric illustration of the Curie principle may be the superposition of two symmetric figures: this gives a figure possessing only those symmetry elements that are common to both figures at a predetermined point in their mutual arrangement. Thus the concept of symmetry is expanding. Symmetry is regarded as a state of space that is characteristic of the environment in which a phenomenon occurs. For example, the following factors should be taken into account during crystal growth: • • •

the status and structure of the environment (e.g., solution or melt); the movement of seed during crystal growth with respect to the environment in which the crystal is forming; and the impact of other physical factors on growing crystals.

The shape of the grown crystal retains only those elements of its own symmetry that coincide with the symmetry of the medium; therefore some of the crystal’s symmetry elements disappear externally. Only those elements of proper symmetry should be accepted that coincide with the symmetry of the environment. Thus the crystal responds to changes in the conditions of crystallization. Therefore different natural shapes of a crystal correspond to crystallization conditions. Under essentially different physical and chemical conditions, the minerals of the same composition acquire different structures (phenomenon of polymorphism). For example, the forms of pure carbon are polymorphic: cubic diamond, hexagonal diamond, multilayer graphite, globular fullerene, quasi-1D carbon, flat graphene, cylindrical nanotubes, etc.

21

22

CHAPTER 1 Structure of electronic materials

Using the known point symmetry of a crystal, the Curie principle allows to predict what physical effects (typical for a given symmetry) can occur. However, the symmetry conditions (that follow from fundamental laws) are considered as not necessary, but rather as sufficient, because of their “abstract” nature for implementation to a physical phenomenon. Direct and inverse piezoelectric effects can occur only in 20 of the 32 possible classes of crystals, each of which is characterized by its symmetry group. These groups comprise a set of symmetry elements: axes of symmetry and planes of symmetry. Crystals with the center of symmetry cannot be piezoelectric. There are 11 such classes of crystals out of 32 (however, there is another class 432 that refers to the noncentrosymmetric classes, but piezoelectricity is not observed in it in spite of the center of symmetry being absent). The 10 polar groups of crystals that exhibit a pyroelectric effect are called the “pyroelectric group” (Fig. 1.15). A common feature of this group is that they lack certain elements of symmetry: the center of symmetry, transverse plane of symmetry, and any axes of symmetry of an infinite number, perpendicular or oblique with respect to the current axis. Physical and crystallographic installations of crystals. To investigate the relationship between crystal properties and crystal symmetry, it is necessary to consider the orientation of a plate sample cut from a single crystal (as well as the orientation of crystalline rods or disks) as to the crystal elements of symmetry. This operation is performed using crystal installation that is determined by the choice of the coordinate system with respect to the symmetry elements of the crystal [8]. As a rule, two settings of crystals are used: the crystallographic installation (used during electronic spectra study in semiconductors and metals) and the physical installation (used in crystal physics including material sciences). In the crystallographic installation, the coordinate axes should be chosen parallel to the directions of space lattice translations. In this case, the crystallographic axes of a coordinate system may not be mutually perpendicular to each other (in crystals belonging to the triclinic, monoclinic, rhombohedral, or hexagonal systems).

FIG. 1.15 Ten polar crystallographic symmetry groups [6].

1.2 Symmetry of crystals

Together with the crystallographic coordinate system that is not orthogonal for all classes of crystals, the orthogonal coordinate system, with axes denoted by either X, Y, and Z, or 1, 2, and 3, is selected to describe the physical properties of crystals. In this case, the symmetry axes or normals to planes of symmetry are chosen as coordinate axes. For example, in monoclinic syngony, the Y-axis is oriented along a single axis of the second order, or along a direction perpendicular to the single plane of symmetry. The remaining two axes X and Z can be chosen arbitrarily, usually by a “binding” to the most advanced face or edge of a crystal. In the orthorhombic system, the crystallographic axis must be directed along the axes of second order, or perpendicular to the plane of symmetry m. In the class mm2, the symmetry axis is defined as axis Z; for a tetragonal crystal, the Z-axis is the axis of the fourth order. In all classes of point symmetry, the X- and Y-axes (except for 4, 4, and 4/m, where they are chosen randomly) are oriented along twofold axes or perpendicular to the plane of symmetry m. In the hexagonal system, the Z-axis is oriented along the axis of symmetry of the highest order. In classes 3m and 6m2, the X-axis must be directed perpendicular to the plane of symmetry. In cubic crystals, the axis 2 is selected as Z-axis (for classes 23 and m3), or 4-axis and 4-axis (for other classes). The X- and Y-axes are oriented along the axes of symmetry. Importantly, in all cases the X-axis and the Y-axis are selected in such a way as to form the right-hand coordinate system. In case of any spatial lattice symmetry, the size of the unit cell (a1, a2, and a3) is selected as a scale (individual segments). Coordinates of any point of crystal are uniquely determined by the direction of the symbol. The crystallographic direction is a direction of line that runs at least two lattice points. One of these points can be taken as the origin: [000]. The crystallographic direction r is completely determined by aligning on it the lattice point closest to the origin, and it is denoted as [mnp], where m, n, and p are the Miller indices. The vector R that coincides with the given direction can be written as: R ¼ ma1 + na2 + pa3 :

Irrespective of the angle between the coordinate axes, the crystallographic axes must follow Miller indices: the X-axis is [1 0 0], the axis Y is [0 1 0], while the Z-axis is [0 0 1]. The indices of axes are written in square brackets. The position (orientation) of each face of a crystal can be described by using the ratio of unit segments a, b, and c to segments A, B, and C that cut off axes by a given face (Fig. 1.16). The set of relations a/A, b/B, and c/C can always be expressed as the ratio of integers a/A:b/ B:c/C ¼ h:k:l. These three numbers h, k, and l determine the position of each edge of the crystal, and they are commonly called Miller indices of edge, written in parentheses as (hkl). In this description, the crystal face is displayed by the position of unit normal n to it, while a set of Miller indices is the component of the normal vector N to a given face relative to the basis of the reciprocal lattice of the crystal: b1, b2, and b3, which is also called the reciprocal basis; that is, N5hb1 + kb2 + lb3 ,

because n1 : n2 : n3 ¼ A : B : C ¼ 1/a : 1/b : 1/c.

23

24

CHAPTER 1 Structure of electronic materials

FIG. 1.16 Explanation of symbols: the position of the plane is uniquely determined by integer intercepts on the crystallographic axes of the coordinates.

Fig. 1.17 shows the main crystallographic directions with an example of a cubic lattice. The reciprocal lattice. This notion is introduced by Gibbs and represents the Fourier transformation of the Bravais lattice that is also known as the direct lattice (that exists in real space). The reciprocal lattice exists in the reciprocal space, also known as the impulse (momentum) space. If one makes a Fourier transformation with the reciprocal lattice, the original direct lattice will be found again, because the two lattices are Fourier transformations of each other [3]. The concept of a reciprocal lattice is used to solve many problems related to wave processes in crystals, for example, in X-ray experimental study or when using electronographic and neutronographic methods of crystal investigation. Moreover, the reciprocal lattice is widely used in the physics of semiconductors and

FIG. 1.17 Marking of main crystallographic directions and planes.

1.2 Symmetry of crystals

metals to describe the motion of electrons in the periodic structure. In that case, the concept of the Brillouin zone is introduced. If a normal direct lattice is based on the translation vectors {a1, a2, a3}, the axes of the reciprocal lattice {b1, b2, b3} are defined as the vector products: b1 ¼ ½a2  a3 ,b2 5½a3  a1 ,b3 ¼ ½a1  a2 :

Furthermore, it is possible to set a reciprocal lattice by the scalar products: ðb1  a1 Þ ¼ ðb2  a2 Þ ¼ ðb3  a3 Þ ¼ 1,

and ðb1  a2 Þ ¼ ðb1  a3 Þ ¼ ðb2  a3 Þ ¼ ðb2  a1 Þ ¼ ðb3  a2 Þ ¼ ðb3  a1 Þ ¼ 0,

inasmuch as b1 ?a2 ,b1 ?a3 ,b2 ?a3 , b2 ?a1 ,b3 ?a2 ,b3 ?a1 :

Thus the absolute value of each of the vectors b1, b2, and b3 is inversely proportional to direct lattice distances: h i h i h i ! ! ! ! ! !
!

!

!
a  a a  a a  a 2 3 3 1 1 2






b 1
¼ ! h! ! i ;
b 2
¼ ! h! ! i ;
b 3
¼ ! h! ! i : a1  a2  a3 a2  a3  a1 a3  a1  a2

Direct and reciprocal lattices are mutually connected, that is, a lattice, built on vectors a1, a2, and a3, is the reciprocal lattice relatively b1, b2, and b3, when |b1 | ¼ ða2 a3  sin αÞ=V,|b2 | ¼ ða3 a1  sin βÞ=V,|b3 | ¼ ða1 a2  sin γ Þ=V,

in which connection: cos α∗ ¼ ð cos β cos γ  cos αÞ= sin β sin γ; cos β∗ ¼ ð cos α cos γ  cos βÞ= sin α sin γ; cos γ ∗ ¼ ð cos α cos β  cos αÞ= sin α sin β,

where V is the volume of a unit cell of the direct lattice, whereas α*, β*, and γ* are angles between the axes of the reciprocal lattice. In physics of crystals, the accurate establishment of rules is very important, that is, rules of orientation of symmetry elements along coordinate axes, because this affects the unambiguous determination of main directions and faces in crystals. The choice of positive directions of axes in crystals is imposed by certain conditions. For example, to study the electrical properties of pyroelectric crystals, the positive direction of the axis (that coincides with polar axis) should be selected as the direction that shows positive electrical charges while heating [7]. The conception of the “reciprocal lattice” is introduced in crystallography mainly to describe the periodic distribution of reflectivity of a crystal relative to X-rays. The reflection of X-rays from planes of crystal is described by the Wulff-Bragg formula 2d sin θ ¼ nλ,

25

26

CHAPTER 1 Structure of electronic materials

where d is the interplanar distance for the family of parallel planes of reflection; θ is an angle, supplementary to the angle of incidence (or angle of reflection, calculated from the atomic plane); n is an integer factor that characterizes the order of the diffraction spectrum; and λ is the wavelength. From the Wulff-Bragg conditions, it follows that, at constant λ, the big d corresponds to a small θ, that is, the greater the interplanar distance, the closer the direction of reflected rays to the direction of the incident beam. The reflection of X-rays from an infinitely extended crystal should be dotty, ideally.

1.3 BASIC STRUCTURES OF CRYSTALS USED IN ELECTRONICS In any crystal structure, a definite group of bound atoms exists that corresponds to the major structural unit—the basis. This is a set of particles, coordinated within an elementary cell; therefore, the whole crystal structure can be obtained by repetition of this basis using translations. An important parameter of a structure is the CN. For a given atom (or ion), this is the number of the nearest (neighboring) atoms or ions in the crystal structure. This number is determined somewhat differently for molecules and crystals. The number of interior atoms is the bulk CN, whereas the number of atoms located on the surface of a crystal is the surface CN. If the centers of the nearest atoms or ions connect with each other by straight lines, it generally gives rise to the coordination polyhedron [8]. The atom, for which the coordination polyhedron is built, is located in the center of the polyhedron (Fig. 1.18). The coordination polyhedron is not related to the outward form of a crystal. The size of the structural unit (atom, molecule, or ion) depends on its location in a particular structure. However, when considering different structures, it is important to compare the sizes of structural elements. In part, the term “atomic radius” is used, but it should be noted that an isolated atom has no certain radius, because its electronic cloud, theoretically, extends to an infinite distance from the nucleus, although

FIG. 1.18 Coordination polyhedra: (A) dumbbell, CN ¼ 2; (B) triangle, CN ¼ 3; (C) tetrahedron CN ¼ 4; and (D) cuboctahedron, CN ¼ 12.

1.3 Basic structures of crystals used in electronics

really it becomes very diffuse at a distance of a few angstroms. The only atomic radius that has certain sense is the Bohr radius of the outer orbit. In case of metals, the radii of the metal ion could be defined by dividing in half the interatomic distance of pure metal. In fact, whenever the atom of any metal is described, three main types of radii are known: ionic (ri), metallic (rm), and covalent (rc). For example, in sodium ri ¼ 0.95, rm ¼ 1.57, and rc ¼ 1.59. It is seen that covalent and metallic radii are very close in size; this is expected because metallic and covalent bonds are related. However, the ionic radius of a cation is smaller, because the outer electrons are removed from the atom, while remaining electrons are located on the levels of internal electron clouds (i.e., located much closer to nucleus). In all nonmetals as well, three radii can be chosen: ionic (ri), covalent (rc), and van der Waals (rv). It should be noted that the typically covalent radius is much smaller than the ionic and van der Waals radii. For example, in oxygen, rc ¼ 0.66, ˚ . This can be explained as follows: suppose that the radius ri ¼ 1.40, and rv ¼ 1.40 A of the atom is determined by the position of the maximum in the radial distribution of the electronic density curve. When the atom attaches electrons and creates an anion, this maximum shifts to a longer distance due to the enlargement of the number of external electrons and the increased screening from the nucleus. Therefore the curve of radial distribution of electronic density shifts toward a larger radius. In ionic crystals, the shells of cations and anions are completely filled with paired electrons; therefore the overlap of such shells is small. However, in covalent crystals, their shells are occupied by paired electrons and therefore the overlap is as large as possible; this interpenetration of electronic orbitals should cause a reduction in the size of the atom. The van der Waals radii, as expected, are very close to ionic radii. In case of a van der Waals connection, as well as for ionic bonding, completely filled electronic shells are situated adjoining each other. As a result of a slight overlap, the interatomic distances get larger values. During consideration of complex structures, the following circumstances must be considered: 1. In ionic structures, the amount of anions surrounding any cation can be determined by the ratio of their radii. When the covalence part of the bond increases, the coordination of the particle will be determined by hybrid orbitals, and the CN should be less than would be expected from an exclusively ionic model. 2. When there is a choice between “ionic” and “layered” structures, in case of covalent compounds the latter is preferred, especially at low temperatures. If hydroxyl groups are present, their hydrogen bonds also tend to stabilize the layered structure. 3. In complex structures, the charges—attributed to atoms in assumption of an ionic model—aspire for compensation (saturation) within inner surroundings. 4. If stoichiometry allows, multivalent cations are located far away from each other; thus each anion is directly associated with only one cation. 5. The cations, after maximal mutual distancing, seek to form linear bridges through anions. This effect is most typical for multivalent cations with a small radius.

27

28

CHAPTER 1 Structure of electronic materials

Next, some typical structures of chemical elements and binary compounds in solids are discussed as examples. Typical structures of metals. Early ideas of the structure of metals lie in the model of “free electronic gas.” According to this model, it is assumed that atoms of a metal are entirely ionized and they are densely packed in the environment of free electrons. With this model (with free electrons and nondirectional bonds), high electrical conductivity and ductility of metals are easily explained. With regard to the structure, electronic gas occupies a small volume (free electron radius is estimated as 1013 cm); therefore, ions of metal form a most densely packing. For example, the structural type of copper corresponds to a dense three-layer packing of identical balls according to a face-centered cubic elementary unit (Fig. 1.19A). This cubic unit cell contains four atoms; its CN ¼ 12, whereas its coordination polyhedron is cuboctahedron. This type of structure is intrinsic for simple metals (gold, silver, nickel, aluminum, etc.) as well as for noble gases in the solid state (e.g., Ne, Ar, Cr, Xe). The balls shown in Fig. 1.19A are not hard: in crystallography, the outer electronic shells of ions, atoms, or molecules are modeled. These electronic shells can be imagined as a “cloud” of electrons that are resilient, but can partly penetrate into each other. This manner of atoms modeled as “hard balls” is a convenient method to describe structures of crystals. The structural type of magnesium is another example of the simple structures of metals (Fig. 1.19B). The arrangement of atoms in this case corresponds to a hexagonal (two-layered) shape with the most dense packing. All atoms in this case are the same, with CN ¼ 12. Ideally, in a densely packed hexagonal metal, the ratio of the unit cell height to the distance between neighboring atoms is c/a ¼ 1.633, although c and a are different in various metals. The structural type of magnesium is typical of many metals: Be, Cd, Mg, Ni, Zn, etc. In addition, there are other structures of metals (not shown in the figure), namely, the α-tungsten structure. This is a space-centered cubic structure, in which the unit cell contains two atoms. The CN of such a structure is CN ¼ 8, while coordination polyhedron is a cube. A structure of this type is typical for some metals: Ba, Cr, α-Fe, K, Li, Mo, Na, and others.

FIG. 1.19 Location ions in the structures: (A) copper and (B) magnesium.

1.3 Basic structures of crystals used in electronics

FIG. 1.20 Structural typing of diamond (A) and graphite (B).

Basic structures of semiconductors. The structure of the diamond that was previously shown schematically in Fig. 1.5B is shown in greater detail in Fig. 1.20A. This structure is characterized by the manner in which atoms occupy face cells of two units that are inserted into each other, offset by 1/4 along the diagonal of the cubic unit cell. The structure of the diamond is peculiar to materials with sp3-hybridization of atomic orbitals. Each atom forms four bonds with its neighbors. The basic structural unit cell of the diamond contains eight atoms, with CN ¼ 4, while coordination polyhedron is regular tetrahedron. The density of structural packing in diamond is much lower than in others. Germanium, silicon, and gray tin also have structures similar to that of the diamond. Similar to this structure is the structure of zinc blende, that is, if two diamond sublattices are occupied by different types of atoms, such as in crystals ZnS or GaAs. The structure of graphite is characterized by a cleavage. In the hexagonal modification (Fig. 1.20B), graphite layers are placed such that atoms of third layer are located just above the atoms of the first layer at a distance far greater than the distance between the atoms inside the layer. The unit cell contains four atoms, with CN ¼ 3, and a coordination polyhedron that is an equilateral triangle with a central atom that is located slightly above or below the plane of the triangle. Like other layered structures, graphite has some polytypic modifications. The sphalerite and wurtzite structures. Crystals of ZnS are crystallized in a cubic sphalerite structure (also called the zinc blende structure), or they are crystallized in a hexagonal wurtzite structure. In both structures, each zinc ion is tetrahedrally surrounded by sulfur ions; in turn, each ion of sulfur is surrounded by ions of zinc. This structure should be seen as densely packed with sulfur ions and in which the zinc ions occupy half of the tetrahedral voids. Accordingly, the structure of sphalerite has cubic packing; therefore this structure resembles a diamond, in which the unit cell

29

30

CHAPTER 1 Structure of electronic materials

FIG. 1.21 Structure of sodium chloride: (A) centers of ion location, (B) tetrahedral surrounding of ions.

contains four sulfur anions and four zinc cations with CN ¼ 4. The wurtzite structure has a hexagonal packaging of the unit cell that is schematically shown in Fig. 1.21C. The CN is 4 in both structures. The sphalerite and wurtzite structures are characteristic of many semiconductor crystals: AgI, AlR, GaR, GaAs, CdS, CdTe, SuI, SuF, NgS, NgSe, ZnS, ZnSe, ZnTe, ZnO, etc. There are certain differences between diamond-type semiconductors and metals. In semiconductor compounds, each bond has a pair of electrons, and these electrons are localized with these bonds. Metals are characterized by a higher number of bonds than the number of electron pairs, whereas the electrons are not localized, but “smeared” throughout a structure. The basic structures of dielectrics. The structure of rock salt (otherwise halite, sodium chloride, or NaCl) is shown in Fig. 1.21. Chlorine anions form a facecentered cubic structure while sodium cations fill all octahedral voids. The unit cell contains four sodium ions and four chloride ions; the CN ¼ 6; the coordination polyhedron—an octahedron—is shown in Fig. 1.21B. The structure of halite is characteristic of alkali halide crystals (except cesium haloids) and some oxides of transition metals (MnO, FeO, etc.), as well as for nitrides and carbides of transition groups Ti and V, haloids of silver (AgCl, AgBr, AgF), tin sulfides, and selenides. The structure of cesium chloride is characterized by anions of chlorine that occupy the cubic cell, whereas cesium cations retain voids between them. In the compound CsCl, the radius of the Cs+ cation is 1.69 A while anion Cl has a radius of 1.81 A. Smaller Cs+ ions should determine the CN; its ionic radii ratio is 0.93 and CN ¼ 8, which is observed in reality. The coordination polyhedron is cubic; therefore the structure of CsCl is cubic space-centered (Fig. 1.22). It should be emphasized that these (and many other) images of crystal structures represent only the spatial arrangement of the centers of atoms (middle position of

1.3 Basic structures of crystals used in electronics

FIG. 1.22 Cubic structure of cesium chloride (A) and fluorite (B).

their nuclei). However, the maximum electronic density is located around nuclei or along areas between adjacent cores. The CsCl type of structure is peculiar for some alkali halide crystals (CsBr, CsI, RbCl, RbBr, RbI, TlCl) and for some metal alloys (FeTi, NiTi, CdAg, AgLi). The fluorite (CaF2) structure is characteristic in some compounds of the ABII type (Fig. 1.21B). Preserving conditions of neutrality results in the fact that the CNs of cations and anions are different. The calcium ion is surrounded by eight ions of fluoride whereas each fluoride ion is surrounded by four calcium ions. This structure is most favorable for the emergence of Coulomb interaction forces between particles. Compounds that crystallize in the fluorite structural type are SrF2, ZrO2, Li2O, CuF2, K2O, CeO2,. Cu2Se, Na2O, etc. ˚ while The rutile (TiO2) structure. The radius of the Ti4+ cation equals 0.68 A 2 ˚ radius of the anion O is 1.40 A. The ratio of the radii is 0.49; therefore, around the titanium ion, there are six anions of oxygen. Thus each titanium atom is surrounded by the octahedral group of O2, and each oxygen ion is surrounded by three Ti4+ cations, thereby forming a triangle (Fig. 1.23A). Crystals that belong to the structural group of rutile are MgF2, MnО2, CгО2, гPbO2, ZnF2, etc. The structure of corundum (Al2O3) is typical of some sesquialter oxides, such as Fe2O3 (hematite), Ti2O3, and Cr2O3. In corundum, each aluminum atom is surrounded by distorted octahedral groups of oxygen atoms. In accordance with the requirements of preserving neutrality, every oxygen atom is surrounded by a tetrahedral group of aluminum atoms. Details of the structure of corundum are shown in Fig. 1.23B: oxygen atoms form a hexagonal densely packed structure that comprises the layers O and A1, whereas one-third of the possible locations in aluminum remain unoccupied. Moreover, there are other, less stable forms of A12O3, demonstrating the “flexibility” of some structures. One of A12O3 modifications crystallizes in the NaCl structural type; however, this structure is faulty: of three cationic places, one remains free. Another A12O3 modification crystallizes in the structural type of spinel.

31

32

CHAPTER 1 Structure of electronic materials

FIG. 1.23 Structures of rutile (A) and corundum (B).

The spinel structure is peculiar to many oxides with the formula Me3O4. In this structure, oxygen atoms form a dense cubic packing. The structure is formed by connected structural units (Fig. 1.24). Panel a in Fig. 1.24 corresponds to a case where metallic ions are found in octahedral sites, whereas panel b shows metallic ions occupying the tetrahedral sites. In addition, there are some vacant positions. Typical representatives of the spinel structure are MgAl2O4 as well as magnetite (Fe3O4), hercynite (FeAl2O4), and chromite (FeCrO4). It is possible to assume that they are all folded by one two-valence cation, two trivalent cations, and by four oxygen anions. The correspondent unit cell contains 32 oxygen ions, which is 8 times more than that specified in the formula. This oxygen skeleton is completed by 8 of 64 possible tetrahedral and 16 of 32 possible positions of octahedral cations. In some spinels, the tetrahedral positions are occupied by two-valence ions.

FIG. 1.24 Spinel structure: (A) octahedral coordination in Al, (B) tetrahedral coordination in Mg.

1.3 Basic structures of crystals used in electronics

FIG. 1.25 Perovskite structure.

The structure of perovskite (CaTiO3) is peculiar to orthorhombic systems, wherein Ca2+ ions (with CN ¼ 12) are found inside cuboctahedral cavities, created by octahedrons that are connected to each other by the vertices, whereas Ti4+ ions are found inside the octahedron. Many compounds with octahedral location of cations form a perovskite structure. Typical representatives of such compounds are ferroelectrics of BaTiO3 type, potassium iodate (KIO3), and potassium-nickel fluoride (KNiF3). The structure of perovskite is shown in Fig. 1.25. At first glance, it is in some respects similar to the structure of cesium chloride, but with one important distinction: each oxygen atom in the group (TiO6)8 is located between two titanium atoms; therefore the linkage Ti-O-Ti is linear. Thus it is possible to assume that the structure of perovskite is created from groups of TiO6, bound in the vertices, and Ca2+ is located above the center of each face of the octahedral group. A general charge compensation must occur in the structures within inner surroundings. Therefore a structure in which charge compensation could take place only at large distances cannot be steady. In other words, large structural units should not have a residual charge, whereas paired electrons of covalent bonds must not move on long distances. The structures of molecular crystals. When the particles creating a crystal are whole molecules, they are associated in the crystal by intermolecular forces. Because these forces are many times weaker than forces that bind particles in the ionic, atomic, or metal crystals, molecular crystals have a low hardness, low melting point, and significant volatility. Inert gas crystals have the simplest cubic structure. Although their lattice is formed by atoms of inert gas, the nature of bonds relates to molecular structure, as the valence force plays no role in the formation of these crystals. Due to the

33

34

CHAPTER 1 Structure of electronic materials

spherical shape and spherical symmetry of the interacting atoms, the crystals of inert gases energetically form the most advantageous structure: a face-centered cubic lattice that is characterized by very dense packing of atoms. Substances made of diatomic molecules usually form crystals of a more complex structure. Especially complicated structures are formed in materials containing polyatomic molecules. Only the most symmetric and relatively simple molecules such as CH4, CBr4, and so on crystallize in a cubic system. The tetrahedral angle between them equals 109°280 , which corresponds to the lowest energy of electron repulsion. The most common molecular crystal is ice (H2O). Its crystalline structure resembles the structure of a diamond because each molecule is surrounded by the four closest molecules, which are located at the same distance and are placed at the vertices of a regular tetrahedron whose angles are 109°280 (Fig. 1.26). Due to a small CN, the structure of ice looks like netting that shows low density. At present, there are 14 known crystalline modifications of ice, but the most common is the hexagonal structure. In ice, the hydrogen bonds that form between water molecules play an important role. Each oxygen atom is surrounded by four other oxygen atoms, linked via the hydrogen atom. Two of four hydrogen atoms are located closer to the given oxygen atom, and they create a water molecule. Two others are attached with this molecule by hydrogen bonds, and they are a part of other water molecules. The distance between two nuclei of oxygen atoms of neighboring mol˚. ecules is 2.76 A Such an arrangement is very far from the dense stacking of molecules: in such ˚ ), the molar volume of ice would be cases (when packing balls of radius 1.38 A approximately two times smaller, because when molecules order in the more dense structure, their mutual orientation cannot be stored, but it is necessary for the emergence of hydrogen bonds. The distortion of a bond’s angles requires considerable

FIG. 1.26 Model of ice.

1.3 Basic structures of crystals used in electronics

energy expenditure. This explains the friable structure of ice and why the density of ice is less than that of water. With modifications too, ice is as crystalline and as amorphous a structure, differing by the relative positions of water molecules and different properties. After ice melts, the ice water partially preserves the structure of ice. Many types of molecular crystals including, particularly, macromolecular compounds (polymers) are complex substances with molecules of high molecular weight. They are constructed from a large number of elementary units that are repeated. Polymers are formed by the interaction of identical simple molecules—monomers. These compounds are rubber, artificial fiber, plastics, cellulose, protein, etc. In their properties, the macromolecular compounds resemble colloids because the dimensions of macromolecules are close to the size of colloidal particles. Most polymers have no crystalline structure (polystyrene, polyvinylacetate, rubber, etc.). However, there are some polymers with pronounced crystalline structure, such as polydiacetylene. Organic substances mainly consist of molecules that have stable structure, such that they can form crystals. Moreover, the concept of intermolecular radii and compact packaging can be introduced for molecular crystals, taking into account the characteristics and geometrical structure of molecules. Molecular crystalline structures tend to have lower symmetry than inorganic structures. Some macromolecular compounds have such structures wherein crystalline regions alternate with amorphous ones. For example, in natural cellulose, 70% of molecules are well ordered, while in 30% are disordered. The properties of polymers can change quite widely, depending on their molecular and supramolecular (crystalline or amorphous) structure, and they thus find different applications in practice. The solid solutions. A crystal or polycrystal can consist of several components (e.g., two components in metals that are alloys). Usually, these components cannot chemically interact with each other (forming compound) but have the ability to be mutually dissolved (as liquid, but in the crystal state), forming so-called solid solutions (or mixed crystals). Here, atoms of one element are introduced into another lattice, creating solid solutions of intrusion or solid solutions of substitution. The solid solutions of intrusion arise when atoms of an element that dissolves are placed in an empty space in the lattice of solvent. Obviously, the size of atoms of the element that dissolves must be smaller. Usually, it should be less than 0.63 of solvent atom size because, if it is larger, there might be a distortion of the lattice. The solid solutions of the substitution are formed by partial substitution of atoms of the solvent by atoms that dissolve. This process can occur without incurring significant stresses in a lattice only in such cases where the size of atoms does not differ greatly. Both types of atoms must be sufficiently close in their chemical properties, and it would be the best if they belong to similar subgroups of the periodic table. Polytypicism is a property of such structures that are built of identical structural elements but have a different sequence of their location. In the plane layers, the structure of the polytypic lattice usually remains unchanged; however, in directions perpendicular to the layers, the lattice parameters are different, even though they multiple distance between adjacent layers. The phenomenon of polytypicism is usually seen

35

36

CHAPTER 1 Structure of electronic materials

with densely packed but layered structures. It is associated with a difference in the relative orientation of atoms and results in a change in their identity period. An example of a compound in which a large number of polytypic structures is found is the semiconductor silicon carbide (SiC). This crystal exists as in the sphalerite cubic modification—in hexagonal modification. The simplest structure for silicon carbide is the six-layer packaging (n ¼ 6). However, there are other polytypic structures of SiC wherein n ¼ 4; 15; 21; 33; 65; 192; 270; 400; 594; or 1200. Another semiconductor crystal—ZnS—has approximately 10 modifications. Moreover, polytypic structures are observed in graphite, molybdenite, and other crystals. Polytypicism significantly affects the physical properties of crystals, especially their optical properties. Isomorphism is a property of chemically closed atoms, ions, or other structural elements to replace each other in a crystal lattice and form a continuously variable composition. Here, atoms with the same valences, bonding type, and polarization that are similar in size (with deviation of not more than 5%–7%) are chemically closed. Isomorphic substances with close but not identical composition crystallize in a similar form. Both germanium and silicon crystals are examples of isomorphism. The density, lattice parameters, and hardness in an isomorphic row of mixed crystals Ge-Si vary linearly. However, as the energy spectra of germanium and silicon are different, the electron’s energy bandgap, specific conductivity, and thermoelectromotive power in this series of Ge-Si semiconductors vary nonlinearly. By selecting different isomorphic compositions, it is possible to vary the range of operating temperatures and electrical characteristics of semiconductor compounds. Crystalline touchstring can cause crystallization and ordering of another isomorphic substance from a supersaturated solution or melt. The ability of isomorphic substances for mutual growth is used in crystal growth technology. Polymorphism is the property of certain substances to exist in multiple crystalline phases, differing in symmetry of structure and in physical properties. Polymorphic modifications are called allotropic elements. At conformable physicochemical conditions, polymorphic modifications can form stable phases. Each of these phases is stable at a fixed range of temperatures and pressures, and is called the polymorphic modification. The relative stability of different phases is determined by the value of free energy and external conditions. Basically, polymorphic modifications differ in their structure, sometimes by the type of chemical bonds. The change in environmental conditions might influence polymorphous transformation. During these transformations (usually, phase transitions of type I), the release or absorption of heat is seen, as well as the jumps of internal energy and entropy. Thus an abrupt change of many physical properties is observed that depends on the arrangement of atoms in a structure: density, specific heat, thermal conductivity, electrical conductivity, etc. In addition, there are other types of polymorphic modifications that differ very little in their physical properties. The polymorphic transitions between such phases characterize the phase transitions of type II, and generally are described as the “order-disorder” phase transitions.

1.4 Lattice defects in crystals

1.4 LATTICE DEFECTS IN CRYSTALS In solid-state theory, in its first approximation, it is always assumed that the structure of crystals is ideal, that is, the location of atoms in unit cells as well as around all crystals is strictly periodic. However, in practice in real crystals, these ideally perfect structures are impossible. Defect formation. Defects in the crystals are formed, for instance, during their growth under the influence of thermal, mechanical, and electrical fields (technological defects), as well as under crystal irradiation by neutrons, electrons, X-rays, and ultraviolet radiation (radiation defects). There are point defects (zero-dimensional), linear defects (one-dimensional; 1D), planar defects (2D), and volume defects (3D). In case of a 1D defect, its size in one direction is much larger than the distance between neighboring atoms (lattice parameter) whereas in the other two directions, the size of the defect has the order of a lattice parameter. In a 2D defect, its size in two directions is much larger than the distance between the nearest atoms, and so on. Mechanisms of defect appearance may be quite various. For example, Fig. 1.27 demonstrates a possible mechanism of crystal growth [9]. Atoms are relatively weakly linked to a flat surface of ideal crystal (Fig. 1.27A) but would have better connected near a step formed by two planes (Fig. 1.27B). It is obvious that the atom will be strongly linked in the corner formed by two steps (Fig. 1.27C): this mechanism of crystal growth seems more likely. However, crystal growth becomes even

FIG. 1.27 Possible mechanisms of crystal growth.

37

38

CHAPTER 1 Structure of electronic materials

easier if it happens through screw dislocation (Fig. 1.27D). In case of such a structure, the new atoms easily add and form an endless spiral around the disturbance. Crystal growth in this way is much faster, because it does not require the formation of a new embryo, as in the cases shown in Fig. 1.27A and B. Defect formation in a crystal occurs because of different reasons. First, atoms sometimes leave their ideal position in the crystal lattice, and therefore create defects of structure through thermal fluctuations. With this mechanism of defect formation, a small part of the crystal’s own atoms lose their regular places in a lattice (they become the vacancies), or squeeze among other regular atoms, creating interstitial atoms; in both cases, the ideal structure of a crystal becomes locally broken. Second, defects can be formed due to other reasons. For example, some atoms come out of the crystal surface; this is the simplest mechanism of point-defect (vacancies) formation near the surface of a crystal. Thus these vacancies— unoccupied sites in crystal lattice—are not accompanied by interstitial atoms. Vacancies of this type are called Schottky defects. In most crystals, the energy of vacancy formation is close to 1 eV. In case of defect formation by the Frenkel mechanism, interstitial atoms or ions arise inside a lattice. Due to thermal fluctuations or power external influence (e.g., bombardment of crystal by ions), the foreign atom (or ion) can take root in the regular crystal structure from outside and create “extra” interstitial atoms. It is this very method of introducing foreign atoms into a crystal lattice that is used in modern electronics technology: the point is that a semiconductor should be obviously doped with impurities, whose atoms not only have different sizes but also can have different valences. After annealing, these foreign atoms replace the atoms in the crystal lattice, forming a solid solution. Thus the nonideal defect structure can be planned specially by technological means. The need for defect management in structures (quite necessary for semiconductors) is due to fact that defects substantially affect such parameters of crystals as conductivity, dielectric and magnetic energy losses, electrical strength, and other properties of semiconductors, magnetics, and dielectrics, as well as strongly affect the mechanical parameters (strength) of metals. Therefore many properties of solids are structurally sensitive. However, some other properties (e.g., density, specific heat, elastic characteristics) are only slightly dependent on the presence of defects. These properties are structure-insensitive, being determined, first of all, by the nature of fundamental atomic bonding as well as by crystal chemical composition. Defects are very diverse. Sometimes, they are associated with one another, and it is difficult even to assign them to a definite class. However, it is possible to divide the main types of structural defects according to their dimension. The zero-dimensional (point) defects are characterized by structure violation in the nodes or interstitials of the crystal lattice. These defects are caused, primarily, by the disordered location of main atoms in the crystal. Point (zero-dimensional) defects include all defects that are due to the displacement or replacement of individual atom

1.4 Lattice defects in crystals

(or small group of atoms). They arise in the process of crystal growth, but might also be a result of radiation exposure. Moreover, point defects may be made by implantation; these types of defects are most studied, including their motion, interaction, annihilation, or evaporation. These defects include: • • • • •

vacancies—free, unoccupied by atom lattice point; impurity atoms—replacing one type of atom by another type of atom by substitution in a lattice; intrusion of impurity atom into the interstitial space of the crystal lattice; Frenkel pair—vacancy together with interstitial atoms; and Schottky defect—vacancy arising due to the release of an atom on the surface.

Usually, Schottky defects are seen in ionic crystals as a pair of cationic and anionic vacancies. This defect is often found in the alkali halide crystals. The presence of Schottky defects decreases crystal density, as atoms that create vacancies diffuse to a surface (Fig. 1.28). The defects generated by Frenkel’s mechanism are usually vacancies and interstitial atoms. These defects are typical, for example, for ionic crystals of silver halides where superionic conductivity exists. Vacancy and interstitial atoms can move within a crystal lattice by the influence of thermal movement. Furthermore, Frenkel’s defects are easily formed in structures of diamond-type crystals (silicon and germanium). These defects do not change crystal density. In general, crystal can have both Frenkel’s and Schottky’s defects; thus those that dominate that formation require less energy. In ionic crystals formed by two kinds of particles (positive and negative), point defects occur in pairs. Two vacancies of the opposite sign usually form Schottky’s defect. The pair consisting of interstitial ions and the vacancies left by them is usually the Frenkel’s defect. As already noted, the simplest zero-dimensional defects in crystals are the vacancies and interstitial atoms

FIG. 1.28 Scheme of Schottky defect formation: (A) atom going out from crystal surface; and (B) shifting another atom onto empty position of first atom [9].

39

40

CHAPTER 1 Structure of electronic materials

FIG. 1.29 Schematic representation of interstitial atom (A), vacancy (B), and impurity atom (C).

(Fig. 1.29). The displacement of atoms or ions (point defects) causes deformation and elastic fields around defects [9]. According to classification, except for proper point defects, other types of defects are possible, namely, the impurity defects (Fig. 1.29C): if the size of a defect atom or ion is different from the main atoms of crystal. Such defects, for example, are donor or acceptor impurities in semiconductors; similar are the impurities introduced in semiconductors to form the centers of recombination, the charge-carrier-scattering centers, etc. Around such defects, there arise local tension and distortion of the crystal lattice (Fig. 1.29A and B). If the crystal is ionic, the vacancies in it lead not only to lattice distortion, but also to the appearance of effective charges with a sign, opposite to the sign of the charged ion that is missed. However, during defect formation in crystals, the principle of electroneutrality is efficient. The electrical interaction is very large and therefore the sum of all charges of defects generated in a crystal should be equal to zero: X ni qi ¼ 0,

where ni is the concentration and qi is the charge of the originated defects. Thus, for example, the displacement of an ion from the lattice site into the interstitial position is accompanied not only by charged ion appearance in the interstitial space, but also by charged vacancy in the crystal lattice. As with Schottky’s defects, Frenkel’s defects in ionic crystals provide local electroneutrality. In the atomic crystals (doped semiconductors), the compensating charges appear due to electrons. The introduction of impurity atoms in semiconductors results in the appearance of donor and acceptor centers. The donor center is caused by such an impurity that the valence is higher as compared to the basic atom of the crystal. These centers provide additional electrons in the conduction band of a crystal. Donor centers in silicon or germanium crystals very often are formed by phosphorus atoms. The host atom in a lattice has four valence electrons (Si+4). The replacement of some silicon atoms in crystal by phosphorus atoms (P+5, donor impurity) results in “extra”

1.4 Lattice defects in crystals

electron ( e) as the compensating charge that provides common electroneutrality in a crystal. Thus the positively charged (+ e) ion of a donor generates an electron with a negative charge ( e). The acceptor centers are created in silicon (or germanium) by impurities where the valence is one less than the valence of basic crystal atoms. For example, such an impurity is boron (B3+). Thus when fixed in a lattice, the (immovable) impurity ion has a negative charge ( e). The lack of one electron is seen as a hole (+ e), that is, the mobile positive charge. Therefore the acceptor type of impurity is a lower-valence atom than the own atoms of the crystal, and it gives rise to holes in the valence band. The polaron is a charge carrier, partially bound in a crystal lattice (most often, this is a bound electron or hole). The polaron is not a “static” defect in the crystal, being much more mobile than the vacancies or interstitial ions. However, a polaron is much less mobile than an electron or a hole. As a rule, polarons are peculiar to ionic crystals wherein, under the influence of thermal motion or irradiation, some electrons (and holes) appear. In ionic crystals, the appearance of local deformation of ionic lattice (i.e., local polarization of lattice) is energetically favorable for electron. Thus the electrical field of the electron (or hole) is partially screened by the polarization that reduces the electrostatic energy of an electron (hole). Being a mobile charged formation, a polaron cannot be fully considered a “point defect,” but as a special state of conductive electron in the ionic crystal. The excitons can also be interpreted as mobile point defects in crystals. The presence of excitons is, as a rule, a characteristic feature of semiconductors and dielectrics. In case of exciton appearance, ions (or atoms) in a crystal do not change their location, but become significantly different from their neighbors by infringement of its electronic state. Such a “defect” is Frenkel’s exciton. Because the excited state can be found in any ion, and there is strong interaction between the outer electronic shells of ions, the energy infringement can be transmitted from one ion to another. Therefore moving Frenkel’s exciton in a crystal is not related to the change of ion positions; thus it has (as polaron) a much higher mobility than vacancies, interstitial atoms, and impurities of replacement. In general, the exciton cannot be fully considered a localized defect. The diffusion. In processes of semiconductor device technology, a heterogeneous distribution of donors or acceptors is usually necessary to create the p-n junctions for diodes or transistors. In addition, during semiconductor device operation, the heterogeneous (in space and in time) distribution of charge carriers often arises. Whenever there is nonuniform concentration, the phenomenon of diffusion takes place, and it often plays an important role in the given situation. Therefore diffusion has received a great deal of attention in semiconductor research [9]. Diffusion is the directional movement of molecules, atoms, or charge carries from a region of high concentration to a region of low concentration. Diffusion is caused by the aspiration of any system to reach their equilibrium state, that is, in this case, a leveling of concentration. In the first case (technological), it is the smoothing of admixture additive concentration; in the second case (electronic device operation), it is the decrease of excess concentration of charge carriers.

41

42

CHAPTER 1 Structure of electronic materials

FIG. 1.30 Time dependence of concentration clot in a one-dimensional model of diffusion: (A) degradation with time without external influence; and (B) charge carrier diffusion and simultaneous drift with time in the electrical field (t0 < t1 < t2).

In Fig. 1.30A, in a local area with a coordinate x0 at time t0, an excessive concentration of particles nmax is created. This is a nonequilibrium state; therefore, over time (t1, t2, …) under the influence of thermal chaotic motion, the maximum concentration decreases and the area of increased concentration becomes blurred, aiming for full alignment. In cases when donors or acceptors are locally imbedded in a semiconductor, rather high temperature is needed to smooth down their concentration, and this is an important stage in semiconductor device technology. (At room temperature, a great amount of time is necessary to change the position of admixture additives in the crystal lattice.) In many semiconductor devices, the locally increased concentration of charge carriers is created (usually by injecting of carriers into a specimen from an external circuit). It is obvious that such a concentration of charge carriers is the nonequilibrium state. Therefore electrical current flows from both edges of concentration peak, as shown in a 1D model (Fig. 1.30A). This flow, in which the concentration n(t,x) changes rapidly, is called the charge carrier diffusion. The effect of diffusion is eventually to bring the concentration of charge carriers toward their equilibrium situation wherein their concentration is uniform throughout. As time changes, the peak spreads out in both directions and decreases, although the center of the peak remains at the same place x0. A quantitative consideration of these processes reduces to the basic law of diffusion: Fick’s law, which specifies that, in nonuniform concentration, the density of particle flow j0 (i.e., the number of particles, crossing unit area per unit time) is given by j0 ¼ D  ∂n=∂x,

where D is a constant called the diffusion coefficient. This law states that the flow of diffusion is proportional to the gradient of concentration. Thus the more rapidly n varies, the larger is the flow. The negative sign in a given formula is introduced

1.4 Lattice defects in crystals

for convenience in order to make parameter D a positive quantity. As seen from this equation, j0 is opposite to dn/dx. Fick’s law is valid whether particles are neutral or charged. In semiconductors, where moving particles are charged carriers, the flow j0 is proportional to the electrical current: j ¼ e  j0 , because, to obtain the value of the electrical current, one needs to multiply j0 by the charge of the carrier. After the creation of a clot with increased concentration of charge carriers at time t0, if the gradient force field is switched in (usually, it is an electrical field), the clot of particles will diffuse as before, and the center of the clot will also drift because the applied electrical force influences these particles (Fig. 1.35B; this case can be applied also to the movement of defects under a thermal gradient influence). The 1D defects—dislocations—are crystallographic defects or any irregularity within a crystal structure. The presence of dislocations strongly influences many properties of solids. Dislocation may, furthermore, be interpreted as the linear boundary of a structural violation in a crystal. Mathematically, dislocations can be defined as a type of topological defect, sometimes called the soliton. In other words, dislocations are such violations of crystal structure that have greater length (up to macroscopic size), but their lateral dimensions do not exceed several interatomic distances. Therefore, 1D (linear) defects are defects that, in one direction, are much larger than the crystal lattice parameter, whereas, in two other directions, they can be compared to it. There are two primary types of dislocations: edge and screw. Mixed dislocations are the intermediate cases between these two. The edge dislocation is the border of “excess” atomic plane that splits the crystal. It corresponds to a row of convergent atoms along the end of an additional plane of atoms within the crystal. Fig. 1.31A shows the atom arrangement around edge dislocation whereas the right panels of the figure (b, c, d) demonstrate the possible movement of such dislocations in a crystal. In other words, the edge dislocation is a defect when an extra half-plane of atoms is introduced, distorting the nearby planes of atoms [9]. If an external force is applied

FIG. 1.31 Edge (linear) dislocation (A) and its displacement to left side from one part of the crystal in relation to another (B, C, and D).

43

44

CHAPTER 1 Structure of electronic materials

to one side of the crystal, this extra plane will pass through the planes of atoms, breaking and joining bonds with them, until it reaches the crystal (or grain) boundary. Dislocation is described by two parameters: the line direction (i.e., the direction of running along the bottom of the extra half plane) and the Burgers vector that describes the magnitude of distortion. In case of edge dislocation, the Burgers vector is perpendicular to the line direction. The formation of dislocations in a semiconductor crystal occurs in a process of crystal growth because crystal cooling is not uniform (the surface cools faster than the volume). As a result of uneven thermal expansion, tensions appear in the crystal lattice. When temperatures are higher than the temperature of ductility, the stressed state of the lattice can be, to some extent, “removed” due to the formation of linear dislocations. Below this temperature, dislocations in crystal become “frozen.” The screw dislocation is a result of change in atom location in one part of a crystal in comparison with another. It corresponds to the axis of the spiral structure of distortion that is associated with normally parallel planes (Fig. 1.27D). As shown, screw dislocations are formed during crystal growth and then they remain in the structure. It is possible to say that the problem of crystal growth can be solved by the possibility of screw dislocations rising. If the surface plane crosses only part of the way through the crystal and then stops, the boundary of this cut becomes the screw dislocation. It comprises a structure wherein a helical path is traced around a linear defect (dislocation line) in the crystal lattice. In purely screw dislocations, the Burgers vector is parallel to the line direction. As point defects, dislocations can move through the crystal lattice. However, the movement of dislocations is associated with many limitations, because a 1D dislocation should always be a continuous line. There are two main types of dislocation movements: displacement and sliding. The dislocation displacement is due to the addition (or removal) of atoms from the superfluous half-plane that may occur as a result of thermal diffusion. During this sliding, the extra half-plane of dislocation that takes a definite position in the crystal lattice combines with the atomic plane that is located under the plane of sliding, whereas the neighboring atomic plane becomes the extraneous half-plane. This smooth sliding of the dislocation line can be caused by the shear stress applied to the crystal surface. It is well known that a rod of soft metal, after a series of bends and straightening, stops its bending and eventually breaks. This is the example of strain hardening. At each bending, many new dislocations occur in a metal; when their number becomes so large that they cannot move, the crystal loses its ability for plastic deformation and it breaks on any further impact on a crystal. The 2D and 3D defects. Two-dimensional (planar) defects include intergranular or intercrystallite borders, as well as crystal surfaces and modulated structures inside a crystal. Therefore most common 2D defects are the boundaries between grains that are peculiar for polycrystalline materials. They consist of a large number of singlecrystal grains that are randomly oriented and tightly interconnected. These structures usually are polycrystalline metals as well as dielectric or magnetic ceramics.

1.4 Lattice defects in crystals

FIG. 1.32 Schematic (A) and atomic (B) representation of grain boundaries.

Intergrain borders—interfaces of crystallites—are not necessarily flat surfaces. Boundaries between grains (crystallites) in polycrystalline materials might have significant curvature. Layers of atoms near these boundaries are the areas of the disturbed crystal lattice; thus the thickness of defect layers usually equals several atomic layers, providing smooth transition between disordered regions (Fig. 1.32B). The polycrystalline (block) structure of ceramic materials and metals can significantly affect their electrical, magnetic, and mechanical properties. The surface of a crystal, by its essence, is also a 2D structural defect. Therefore, each real crystal differs from the ideal crystal due to variations in the structure and properties of a surface. The surface is a special state of crystal, with different sized elementary cells that have other symmetry and other energy. Atoms (ions) located on the surface layer are joined by broken chemical bonds which are not saturated. Located on the surface are unpaired electrons of atoms (ions) that tend to form new connections. Most often, the state of the surface is characterized by the bonding of neighboring atoms—either in pairs, or in more complex associations. Surface atoms are combined into larger unit cells, as compared with the volume of a crystal. For example, on the surface, a silicon unit cell has 7  7 atoms (in germanium 2 8 atoms) whereas their fundamental unit cell contains only two atoms. Thus the elasticity of atomic bonds on the surface is changed and, as a result, the characteristic “melting point” is reduced by 10%–30%. Note that crystal growth occurs just from the surface as well as the melting of the crystal, its evaporation and condensation, and the diffusion of atoms deeper into the crystal. The electronic energy spectra of the crystal surface significantly differ from the electronic spectra of the crystal volume. As discussed later, the unique properties of nanocrystalline materials are due exactly to the fact that in nanoparticles (which have a small number of atoms: 10–1000), the ratio of surface-located atoms to volume-located atoms is 90%–20%.

45

46

CHAPTER 1 Structure of electronic materials

The physics of modulated structures can be regarded as a boundary area between the physics of nanomaterials and physics of structural defects. In case of nanophysics, the planar (by-layer) modulation of semiconductor structures may result in a specific electronic spectrum—so-called quantum wells. In some types of magnetics, the arrangement of electronic spins in crystals also exhibits periodic complexity in their magnetic ordering, as compared to the usual crystal structure. Modulated structures can be explained as the coexistence of different periodicity in a crystal. The 3D (volumetric) defects. They are, first of all, the clusters of vacancies that form pores and channels; various imbedded defects, such as gas bubbles; accumulation of impurities in a form of sectors and areas of growth. Three-dimensional defects reduce crystal flexibility, affect its elasticity, and strength as well as change the electrical, optical, and magnetic properties of a crystal. Fig. I.2 in the Introduction schematically shows 3D structural defects in polycrystalline materials. Inside of each crystallite, many interplanar structural defects can be observed. Thus 3D defects are solid interstices, liquid or gaseous phases in a crystal, clusters, and other complications with a macroscopic structure. In materials used in electronic technology, 3D defects might also have a fundamental nature, but this case is not considered here. As a result, some conclusions follow: •

•

•

Part of atoms (or ions) of a crystal may be absent in their positions that correspond to the ideal crystal lattice scheme. These defects are vacancies. Foreign (impurity) atoms or ions, replacing basic particles that form a crystal, or inserted between them, can also be seen in crystals. Point defects in a crystal can also be its own atoms or ions that are shifted from their normal positions (internode atoms and ions), In the process of crystal growth, as well as during its plastic deformation and in many other cases, dislocations arise. Dislocations are places with ordered accumulation of impurities. The distribution and behavior of dislocations under external influences determine many important mechanical properties of a crystal, including strength, ductility, and other aspects. In particular, the mobility of dislocation determines the plasticity of crystals; dislocations also cause the appearance of internal stress and fracture of crystals. The problem of plastic (i.e., irreversible) flow of metals can be solved by the prevention of dislocation movements. Dislocations impede the process of magnetization and electrical polarization because of their interaction with domain boundaries movement. Defects in crystals cause elastic deformation of the structure that leads, in turn, to the appearance of internal mechanical stresses. For example, point defects, interacting with dislocations, can increase or decrease the strength of crystals. Defects in crystals affect absorption spectra and luminescence, light scattering in crystal, etc.; such defects change electrical conductivity, thermal conductivity, ferroelectric properties, magnetic properties, etc.

1.5 Structure and symmetry of quasicrystals and nanomaterials

1.5 STRUCTURE AND SYMMETRY OF QUASICRYSTALS AND NANOMATERIALS As described earlier, crystal structure is defined as a system with long-range ordering of particles. If the structure of the crystal unit cell is known, 3D periodicity makes it possible to predict the location of atoms of any other cell and the relative positions of atoms of the entire structure as a whole. This means that crystal has translational symmetry. The structure of the crystal can be described by the displacement of a single unit cell on three basic vectors of translations. Translational symmetry results in regular crystallographic planes in crystal, thus making clearly identified narrow peaks of X-ray scattering. This feature of the X-ray diffraction pattern is the distinguishing characteristic of crystals. Polycrystalline bodies, in their structure, are similar to single crystals, because they are composed of small randomly oriented crystals. During X-ray beam scattering in polycrystals, a conical symmetry is formed that also gives distinct diffraction maxims, which can be used to obtain lattice parameters as in a single crystal. Significant difference occurs in the X-ray spectra of amorphous solids that are characterized by the blurred picture of diffuse X-ray scattering without clearly identified narrow rings. Such solids in their amorphous state do not show strict 3D periodicity. Thus while defining an amorphous structure, the terms “disordered,” “noncrystalline,” “amorphous,” and “glassy” are synonymous. The arrangement of atoms in amorphous solids, however, is not completely random (as it is in gases). The interactions between atoms in an amorphous body are similar to the forces in crystals and, although there is no long-range ordering, the short-range ordering, generally speaking, is preserved [8]. Short-range ordering in the arrangement of atoms is characterized by such parameters as the length and the angles of bonds as well as by the number of their nearest neighbors. It should be noted that in the amorphous state, because of violations of their structure, these options have some statistical dispersion, and their average values may differ slightly from those values in a perfect crystal. The quasicrystals show a new type of symmetry, different from all aforementioned cases. They demonstrate such elements of symmetry that previously were considered as impossible. The translation symmetry of a perfect crystal obeys rigid restrictions as to the order of rotary symmetry axes, which describe the symmetry of a crystal. As shown earlier, the ideal crystal, except with a trivial axis of the first order, can have symmetry axes only of second, third, fourth, and sixth orders. Solely, these axes can provide the parallel transfer of unit cell when it is multiplied to create a crystal. The symmetry of a perfect crystal does not allow the existence of axes of symmetry of the fifth, seventh, or higher orders. Elementary cells that have such axes cannot completely fill even the plane (and, moreover, the volume). Nevertheless, in 1984, for the first time a metallic alloy was discovered with unusual properties: with the axis of symmetry of the fifth order (Dan Shechtman,

47

48

CHAPTER 1 Structure of electronic materials

FIG. 1.33 Models of quasicrystals structures: (A) icosahedron and (B) dodecahedron.

Nobel Prize for 2011). This alloy was obtained by a rapid cooling of molten aluminum-manganese (with the speed of cooling near 106 degrees Kelvin per second). Grains of this alloy have the form of a dodecahedron with rotary symmetry axes of fifth order. As known, symmetry axes of the fifth order have two types of “regular convex polyhedra” (Fig. 1.33): icosahedron and dodecahedron (the existence of these regular convex polyhedra was first noted by Euler). The icosahedron is a regular polyhedron consisting of 20 faces—equilateral triangles—and it has 12 vertices and 30 edges (Fig. 1.33A). The dodecahedron is a polyhedron consisting of 12 faces (pentagons) and it has 30 edges and 20 vertices (Fig. 1.33B). The dodecahedron and icosahedron can be inscribed into one another, similar to a cube and an octahedron. It should be noted that the icosahedron and dodecahedron can be described by identical elements of symmetry, including symmetry axes of the fifth order. As in ideal crystals, the symmetry of the axis of the fifth order is prohibited; therefore the icosahedron and dodecahedron are never used in the translational symmetry of classic crystallography. As mentioned, the diffraction pattern of X-ray scattering for the aluminummanganese alloy shows regular peaks, corresponding to a structure with a rotary symmetry of the fifth order. This diffraction pattern can be formed only when the atomic structure has the axes of symmetry of the fifth order. This means that icosahedral symmetry characterizes not only grains of metal, but the arrangement of atoms in unit cells as well. The presence of different reflexes in the X-ray spectrum shows the special arrangement of atoms in the structure called shechtmanite (a quasiperiodic crystal), whereas the presence of symmetry axes of the order 5 indicates that this material, in the usual sense, cannot be considered a crystal. Some additional research of shechtmanite by methods of electron microscopy confirmed the homogeneity of

1.5 Structure and symmetry of quasicrystals and nanomaterials

this material and the existence of rotational symmetry of the fifth order in the small areas with sizes of a few tens of a nanometer. At present, many alloys of similar structures are discovered and synthesized, and they are called the quasicrystals. For example, the quasicrystals can be obtained by a sudden cooling of molten aluminum, copper, and iron that, during solidification, form grains of the dodecahedron type. In most synthesized quasicrystals, using X-ray diffraction studies, the icosahedral symmetry has been identified with point group of symmetry, inherent to the rotary axis of the fifth order. In addition, other quasicrystals are synthesized with rotary axes of symmetry of 8th, 10th, and 12th orders (all these symmetry axes are prohibited in the translational symmetry of ideal crystals). Quasicrystals usually consist of metal atoms and (sometimes) of silicon, for example, the alloys Al-Li-Cu, Al-Pd-Mn, Zn-Mg-Y, Al-Cu-Co-Si, Al-Ni-Co, and Au-Na-Si. The structure of quasicrystals is characterized by a combination of alternative local symmetry (icosahedral) that is far from ordering, providing sharp peaks in diffraction pattern, observed in experiment. Following the discovery of quasicrystals with fifth-order axis of symmetry, it seems natural to involve the model that can describe structures by regular icosahedrons and dodecahedrons. For example, the icosahedral clusters can be used as a model, consisting of identical solid spheres that represent atoms. The tetrahedral structure can be formed with four closely linked spheres, limiting their planes passing through the centers of spheres. A compound of 20 tetrahedrons creates a small, distorted icosahedron. A similar structure can be obtained by solid sphere wrapping by 12 equidistant areas. However, between 12 peripheral areas, representing atoms, there are gaps that inevitably occur; each atom would be approximately 5% further apart than the distance to the central atom. Compact filling of space by such an icosahedron-type cluster should be quickly broken, that is, the icosahedral packaging cannot spread to the entire crystal [10]. Some structures, which have short-range icosahedral ordering, acquire the term the metal glasses. They are formed by a very rapid cooling (106 K/s) of the melt of some metals. Such structures have only short-range ordering, and, being amorphous, form an X-ray spectrum with broad diffuse maxima. In quasicrystals, however, X-ray peaks are expressed clearly. To explain the spectra of quasicrystals, the presence of icosahedral clusters with regular distortions on borders is supposed, which could provide long-range ordering in the structure and, therefore, create X-ray diffraction patterns with narrow peaks. Therefore, to describe some complex quasicrystals, the structural units containing a few dozen atoms are proposed. However, a problem arises as to the physical nature of appearance and stability of such complex clusters. Furthermore, X-ray and neutrondiffraction methods showed that, in real structures of quasicrystals, only a small fraction of their atoms have an icosahedral environment. Thus, for actually existing long-range ordering, all quasicrystal structures should have some “nontranslating” arrangement. In other words, filling of infinite space by atoms in these structures can be determined by such an algorithm when a long-range

49

50

CHAPTER 1 Structure of electronic materials

order is ensured without full translational symmetry. The lack of translational constraints allows the structure to have a quasicrystalline axis of the fifth order. Orderly arrangement of structural units can provide a positive interference of X-ray waves scattered by atoms in some areas, and the formation of narrow and strong diffraction reflexes. Some ideas as to quasicrystals modeling in 1D and 2D structures are discussed further. Ensuring long-range order in a 1D structure in the absence of translational symmetry is possible in various ways. For example, long-range order in atom distribution can be modeled by the linear chain of atoms with constant interatomic distance “a” that shifts with the next atom on distance Δ ¼ ε  a sin(2πja), where j is the serial number of atom while ε and σ are some numbers. If the number σ is irrational, displacements of atoms are different, even if one considers an endless chain of countable atoms. This 1D structure might have translations. However, the coordinates of all atoms are determined by a definite law, that is, this sequence is totally ordered structure. The lack of translational symmetry in this case is not due to chaotic displacement of atoms (that is typical for amorphous structures), but by imposition of two nondisproportionate periodicity in their arrangement, whereas the ratio of their periods is an irrational number. The lack of random displacements of atoms that leads to the nontranslational arrangement makes an X-ray diffraction pattern, characterized by distinct maxima. Built in such a manner, the chain of atoms is the example of 1D-quasicrystals. This example shows the feasibility of using irrational numbers in constructing models of quasicrystals [10]. The “Penrose mosaic” shown in Fig. 1.34 can be used as a mathematical model of 2D quasicrystals. This structure is fundamentally different from the classic “frozen” form of perfect crystals. R. Penrose developed the algorithm on how to fill an infinite plane with no overlaps and voids by using figures of only two types. These figures that are needed to build the Penrose mosaic are the rhombuses with the same side. The internal angles of wide rhombus equal 72° and 108° and internal corners of “narrow” rhombus are 36° and 144°. A mosaic made of rhombuses can fill all “endless” flat surfaces, but only at an aforementioned selection of special corners of rhombuses. Notably, that ratio of “narrow” rhombuses to “broad” rhombuses is exactly equal to the “golden section” (Golden section is a number (√ 5  1)/2 ¼ 0.618… equals to the ratio of two parts of a whole (Φ and S) that is subject to the following rule: Φ /(Φ + S) ¼ S / Φ.) Because the “golden section” is an irrational number, in the considered mosaic it is impossible to identify any “unit cell” containing a whole number of each type of rhombuses that could fill the plane. Therefore the Penrose mosaic is not a 2D-crystal in the traditional sense, but it is a 2D-quasicrystal. It is important to pay attention to the following facts: First of all, it is essential that the construction of mosaics is realized by defined algorithms, which is why this mosaic is not a random, but ordered, structure. Secondly, when calculating the scattering of X-rays for structure, formed by atoms located in vertices of the Penrose mosaic, it is found that the diffraction pattern

1.5 Structure and symmetry of quasicrystals and nanomaterials

FIG. 1.34 Penrose mosaic as example of two-dimensional quasicrystalline structures.

has a rotary symmetry axis of 10th order. The Penrose filling contains 10 squares with exactly the same orientation. Thirdly, rhombuses of mosaics (with parallel sides) form five families parallel to each other’s lines, intersecting at angles that are multiples of the angle 72°. Thus, the Penrose mosaic has a long-range ordering, providing diffraction pattern of fifth-order rotational symmetry. After the invention of “shechtmanite,” a 3D generalization of the Penrose mosaic is studied that has icosahedral symmetry. Experiments show that in most real quasicrystals, their atoms have nearest neighbors lying in the vertices of a regular dodecahedron. However, the construction of the structure from hard figures with 20 vertices of the dodecahedron by real atoms may engage in no more than eight vertices. Therefore, the first coordination sphere of each atom has a strong volatility. Such structures are characterized by both short-range ordering and long-range ordering (with not usual translating), which can be built only from two types of rhombohedrons. This mosaic is not possible to obtain traditionally by the translations of one elementary cell. It should be noted that the algorithm of 2D-rhombuses or 3D-rhombohedrons in Penrose mosaics consists of several steps, and therefore has alternatives. Although real quasicrystals grow, some failures of its structure are possible because quasicrystals can be formed in regions of their violations. The presence of such “amorphous” inclusions should lead to widening of peaks in X-ray diffraction pattern, as observed in experiments. In addition, an evidence of the presence of disordered local areas is the low conductivity of metal alloys of synthesized quasicrystals.

51

52

CHAPTER 1 Structure of electronic materials

In the melt metals, depending on alloy components, microsymmetry is created correspondingly to features of the electronic structure of ions that coexist in the melt. Microstructure in melts might be quite diverse; they might have axes of symmetry of the fifth order (as well as axes of symmetry of higher orders), that is, such translational symmetries are forbidden in an ideal crystal. The thermal effect on crystal has the symmetry of a sphere; therefore, it contains any element of symmetry (including axes of any order). Usually, quasicrystals are obtained by a sudden cooling of such alloys, where nontranslational symmetry axes are dominating in the microstructure. Therefore, in case of imbalance cooling (by the “heat shock”), primarily the structures of short-range order become stabilized (e.g., with the axis of the fifth order), which is typical of the local electronic structure of a given melt and not forbidden by sphere symmetry (according to the Curie principle). The remaining symmetry elements (in considering the case of the axis of the fifth order) become “frozen” after structure heatstroke, providing a sort of “long-range ordering.” These formed clusters may have enough inner energy to withstand thermal movement and therefore store the elements of symmetry, unusual for perfect (translational) longrange order. In crystals that experience sharp changes of temperature, these unusual elements of symmetry are stored; therefore the properties of such alloys are not traditional. The symmetry of nanomaterials. Nanomaterials exhibit short-range ordering of their atoms. Their relatives are, for instance, well-studied amorphous metal alloys (metal glasses). In such substances, their structure is changed quite significantly, allowing the creation, for example, of ferromagnetics with such magnetic properties that, in principle, cannot be obtained in the materials with long-range ordering of atoms. Topological models of amorphous materials are well developed and are based on the random dense packing of both hard and deformable spheres: this is close to that seen in nanostructures. With regard to inorganic glasses with covalent bonds and random packing of atoms, these structures correspond to the model of a random and continuous grid of atoms. All of the said models are characterized by a set of different-size spheres, randomly packed to the largest density [8]. They differ in the rules of packaging, in the interaction potentials, in the relaxation modes, etc. In many configurations of random dense packing, the crystallographic structural elements are allocated, as well as the noncrystalline packing of clusters that can be illustrated by the Bernal polyhedra (Fig. 1.35). As known, the CN in crystals might be 4, 6, 8, and 12. In the amorphous metallic alloys, the CN for alloys of transition metals with copper remains only close to CN ¼ 12 regardless of the compound (in ideal model CN ¼ 12). In reality, for example, in Ni-Li and Cu-Ti alloys, the average CN is 12.8. In the alloys of rare earth metals and transition metals, as usual, CN ¼ 12; however, in the amorphous alloys, CN generally decreases. For example, in DyFe2 alloy CN ¼ 7.1  1, while in the alloy TbFe2, CN ¼ 8.4  1.8; thus the environment of iron atoms is approximately the same as in the crystal. Thus the short-range ordering in amorphous and in crystalline states of metallic alloys is different.

1.5 Structure and symmetry of quasicrystals and nanomaterials

FIG. 1.35 Models of amorphous structures clusters: 1—tetrahedron; 2—octahedron; 3—trigonal prism with three semioctahedrons; 4—Archimedes’ antiprism with two semioctahedrons; and 5— tetragonal dodecahedron.

Nanomaterials are small particles of matter (clusters), consisting of 10–1000 atoms, and their properties depend on the number of atoms in a cluster as well as on the relative position of atoms. The size of a nanocluster also has an influence on its shape and symmetry [11]. Consider, for example, the cubic symmetry crystal of magnesium oxide (MgO; Fig. 1.36). An important property of nanoparticles is seen: the difference in the outward form of the same material—crystal, microcrystal, and nanoparticle [12]. In this example one can see a resizing change in the shape of a body. When the size is larger than 100  100 nm2, long-range ordering prevails, and MgO crystal has this intrinsic to its cubic form. However, the microcrystal of MgO tends to have a hexagonal shape, whereas the MgO nanosized particle shows a nearly dodecahedron form. Another important example that demonstrates how internal bonding and symmetry influence the properties of materials is of the various forms of carbon. In the periodic table of elements, carbon relates to subgroup 4; the electronic shell of carbon atom has four valence electrons with configuration s2p2, allowing carbon to have valences 4, +2, and +4. The classification of carbon structures is shown in Fig. 1.37. The classic (3D) structures of carbon are diamond and graphite. The diamond is a 3D form of carbon; its

FIG. 1.36 Various forms of MgO structure: 4 nm—nanoparticle; 5  100 nm2—microcrystal; 100  100 nm2—usual crystal.

53

54

CHAPTER 1 Structure of electronic materials

FIG. 1.37 Classification of different forms of carbon.

structure is formed from the electronic state of sp3-hybridization. In the diamond crystal, each carbon atom is surrounded by four others that are in the tetrahedral sites; neighboring atoms are combined together by a strong covalent bond that determines the hardness of the diamond. The distance between the atoms in the diamond is 0.154 nm. In graphite, carbon atoms are connected with each other, thus forming the hexagonal netting, in which each atom has three neighbors. In such a quasi-2D (plane) form of carbon, its structure originates from the state of sp2-hybridization (Fig. 1.38B). The layers of plane nettings of graphite are accommodated one above another. In covalent chemical bond formation, three electrons from each atom take part in creating σ-bonding. The distance between atoms, arranged in hexagonal mesh nodes of graphite, is 0.142 nm—less than in the diamond. Thus neighboring atoms within each layer of graphite are linked by stronger covalent bonds.

FIG. 1.38 Carbon atom location in various structures: (A) diamond, (B) graphite, (C) carbine, (D) fullerene C60, and (E) fullerene C70.

1.5 Structure and symmetry of quasicrystals and nanomaterials

However, these layers fit together by the weak van der Waals forces, in which four electrons are involved. The hexagonal graphite netting is located at a distance of 0.335 nm from each other, that is, the distance between atoms is more than twice that in the layers. This bonding between layers is the π-bond. A large distance between layers determines the weakness of forces that combine layers. This structure—strong segments, poorly linked—constitutes specific properties of graphite, particularly its flexibility that explains a slight sliding of layers relative to each other, as well as the low hardness of graphite and large anisotropy of its properties. The carbine is a linear polymer of carbon that can be obtained in vitro in the form of long chains of carbon atoms, parallel to each other (Fig. 1.38C). The string (linear) structure of carbine is formed by the sp-hybridized carbon atoms. In the very long molecule of carbine, carbon atoms are strongly linked in chains by the triple bond, as well as by the double bonds between them. Carbyne can be obtained in forms of fiber, powder, and films of different structure: disordered long chains, amorphous and quasiamorphous material with microcrystalline inclusions, and bilayer-oriented chains. Crystalline-type samples of carbyne have the shape of plate-form crystals, as well as samples in the form of fiber up to 10 mm in length. The graphene (Nobel Prize for the year 2010) is the plane polymer of carbon: the layer of carbon atoms with a thickness of only one atom is connected by the sp2bonds in the 2D hexagonal crystal lattice. Graphene can be represented as a single plane of graphite, separated from bulk crystal (see Fig. I.3 in Introduction). Graphene is characterized by big mechanical stiffness and large thermal conductivity. The high mobility of charge carriers in the graphene at room temperature makes it a promising material for use in various electronic devices. In particular, graphene can be regarded as an important material for nanoelectronics that allows, in some cases, to replace the silicon in integrated circuits. The fullerenes are molecular compounds belonging to one of the relatively new forms of carbon. They are closed polyhedra composed of carbon atoms that are located on a surface of convex polyhedron (Fig. 1.39D and E). The discovery of fullerenes was also awarded the Nobel Prize. The most stable form of fullerenes is the molecule C60—a polyhedron made of hexagon and pentagon faces. The fullerites are condensed systems consisting of fullerene molecules. In addition, the topical compounds are the fullerides—fullerite crystals doped with alkali metal atoms. Some fullerides exhibit high-temperature superconductivity, for example, in the fulleride-superconductor RbCs2C60, the critical temperature is 33 K. The carbon nanotubes (Fig. 1.39) are lingering cylindrical structures with diameter from one to several tens of nanometers and lengths up to several micrometers. They consist of one or more sheets rolled into a tube hexagonal graphite planes (graphene) and usually terminate in a hemispherical head. There are both metallic and semiconducting carbon nanotubes. Metallic nanotubes well conduct electricity even in near-absolute zero temperatures, whereas in the semiconductor type of nanotubes at temperature close to absolute zero electrical conductivity is nearly zero but increases with a rise in the temperature.

55

56

CHAPTER 1 Structure of electronic materials

FIG. 1.39 Single-walled carbon nanotubes: (A) schematic representation and (B) fullerene-like closed end of the nanotube.

1.6 STRUCTURES OF COMPOSITES AND METAMATERIALS The term “composite” is implied for multicomponent system in which several materials are combined, being different in composition or form, in order to obtain the specific property of the final material. In this case, individual components of a system retain their individuality and properties to such an extent that they exhibit interphase boundary, and operate the achieving the improved properties that are inaccessible to each component separately. Thus, the properties of composite materials are largely related to the geometric arrangement of components. As an important example, the piezoelectric composite materials are considered. These composites are used in underwater sonars, for medical ultrasound diagnostic, in some electronic instruments, etc. In the simplest case, an ultrasound composite receiver can be constructed of piezoelectric and polymer components (Fig. 1.40).

FIG. 1.40 Element of piezoreceiver consisting of piezoelectric rods in polymer.

1.6 Structures of composites and metamaterials

Cylindrical rods made of piezoelectric material occupy a relatively small volume of composite, yet they provide practically the same signal in the receiver as would be obtained from a solid piezoelement, because the rods are hard whereas the polymer is very malleable; therefore the mechanical signal almost entirely acts on the rods. Thus the electrical capacity of composite piezoreceiver is tens of times smaller, because the permittivity of polymer is hundreds of times less than that of the piezoelectric element. Therefore piezoelectric composites are very promising materials because they open the possibility of effective control over their electrical and mechanical parameters. The advantages of such composites are the high coefficient of cohesion, low acoustic impedance (in good agreement with the impedance of water or human tissue), mechanical flexibility combined with low mechanical quality factor. In addition to increased piezoelectric efficiency, some piezoelectric composites can show a magnetoelectrical effect. These composites are composed of magnetostriction ceramics and piezoelectric ceramic and are capable of producing an electrical response (voltage or current) under the influence of an external magnetic field. The classification of various composite structures is proposed by R. Newnham [13]. Properties of composite can be divided into three major effects: the effect of sum, the effect of combination, and the effect of product. 1. The effect of sum. Assume that one of many physical properties of composite and its components are considered. Suppose that component 1 has a property characterized by parameter Y1 while component 2 has parameter Y2. Then, the composite will have some intermediate value of this parameter—a value between Y1 and Y2. In case of a two-component system, the given property is described in the composite with summary function Y*, shown in Fig. 1.41. In case of sum effect, the obtained dependence of the summary parameter from volume fraction of components may be characterized not only by linear dependence, but also might have a concave or convex shape. Thus, and it is very important, that the average value of Y* in composite will never be more than Y1 or less than Y2. This example is commonly used in the microwave range composite material with predetermined permittivity, set within ε* ¼ 5…40. This composite can be

FIG. 1.41 Effect of sum in two-component composite.

57

58

CHAPTER 1 Structure of electronic materials

prepared from ceramic powder of rutile (ε1 ¼ 100) and polymeric polyethylene (ε2 ¼ 2.5). The dielectric permittivity of such composites depends on the volume fraction of ceramics, but it cannot exceed value ε1. 2. The effect of combination. It is assumed that components of the composite are characterized by two different properties: Y and Z. In this case, sometimes, the average value of obtained certain parameter in a composite may exceed the parameters of both components of the composite. The effect of the increase in the output parameter is determined by the ratio Y/Z that depends on both parameters. For example, in some of piezoelectric composites basic properties of components are combined: high piezoelectric modulus of piezoceramics and low permittivity of polymeric matrix (as in Fig. 1.40). As a result, the piezoelectric sensitiveness of composite (that is dependent on ratio of piezoelectric modulus to permittivity) increases substantially. Therefore, this composite has a significant advantage over the properties of components. 3. The effect of product. Besides, such a two-component composite is considered, wherein one of components has a significant property Y, which is absent in the second component. However, in the second component, a quite different property Z is present, which does not have the first component. In this case, it is expected to obtain, in the resulting composite, these brand-new features and this is the effect of a product. For example, based on this concept, the magnetoelectrical ceramic composite material has been developed, consisting of magnetic components with significant magnetostriction effect (CoFe2O4 that is nonpiezoelectric), and the piezoelectric component (BaTiO3, exhibiting no magnetic properties) [14]. Under the action of a magnetic field on composite, cobalt ferrite shows magnetostriction, which is transmitted to the grains of barium titanate as the stress and results in generation of electrical charges (of voltage) due to the piezoelectric effect of BaTiO3. Thus, due to the composite material, the inexpensive ceramic sensors of magnetic field monitoring are elaborated. Metamaterials are composites in which the heterogeneous medium contains inclusions; however, in this case, unlike other types of composite materials, inclusions are miniature, sometimes even nanoscale, radioelements. Due to these inclusions, metamaterials have unique electrophysical and optical properties, caused by the resonant interaction with electromagnetic field. In metamaterials, a very interesting idea is realized: the possibility to obtain the negative refractive index for microwaves or light. In these materials electromagnetic wave, for example, light is not refracted as usual, that is, it deviates not to the right, but to the left at the negative angle (Fig. 1.42A). Therefore these materials are often referred to as materials with negative refraction (negative index materials—NIM) or left-handed materials (LHM). V.G. Veselago, who theoretically predicted the existence of metamaterials, called them “left environments.” In the usual medium, directions of vector of electric field intensity E, vector of intensity of magnetic field H, and wave vector k form the right triplets, that is, they can be described by right-hand fingers (RHM). In contrast, in a metamaterial these vectors form

1.6 Structures of composites and metamaterials

FIG. 1.42 Effect of light refraction in conventional material and metamaterial: (A) direction of rays in medium; (B) orientation of electromagnetic field vectors in ordinary (RHM) material and metamaterial (LHM).

the left-hand triplet (Fig. 1.42B). However, in left-hand material, LHM, the Poynting vector S that shows direction of energy propagation remains in the right triplet. It is well known that electromagnetic waves can propagate similarly as in vacuum (ε ¼ 1, μ ¼ 1) in a dielectric medium with positive permittivity and permeability. Parameters ε and μ are fully defined for each particular material due to one or other atomic or molecular structure. In ordinary materials, these parameters are defined by electrical polarization (displacement of electrical charges with electrical moment formation) and by magnetization (orientation of elementary magnetic moments). Properties of atoms or molecules follow fundamental laws of physics, and they always lead to positive static values of permittivity ε and permeability μ (at that, in most substances μ is close to one). However, there are exceptions—in ranges of frequencies where the own resonant phenomena are observed: this is possible as for polarization so for magnetization. In the first case, when the phase of dielectric “response” (elastic displacement of charges) lags behind the phase of applied field, the response is described by the negative value of ε. A similar process can occur in case of magnetization, causing negative value of μ. Thus, when resonant response occurs, these narrow frequency ranges are characterized by negative ε or μ (i.e., however, it is accompanied by a very large absorption of electromagnetic waves). In case of ionic polarization in dielectrics, their lattice resonance occurs in the frequency range of infrared waves (1013 Hz), while for electronic shells polarization—in range of ultraviolet waves (frequencies >1016 Hz): both these ranges are quite far from the frequency range of metamaterials expected applications. Thus, at first glance, there is no basis for hoping to obtain resonant phenomena in continuous homogeneous medium as in microwaves so in visible optical range.2 2 Note. However, it can be noted that in piezoelectrics, the electromechanical resonance is possible that also leads to negative value of ε. Usually, this resonance occurs at frequencies of 105…107 Hz (depending on size of piezoelement); to realize this resonance at microwaves, the size of piezoresonators should be only a few micrometers. It is obvious that microelectronic technology is responsible for actualizing this case.

59

60

CHAPTER 1 Structure of electronic materials

For this reason, the metamaterial can be obtained exclusively from the noncontinuous and inhomogeneous medium: metamaterials always are the composites [15]. Usually, metamaterials are constructed from the discrete resonant micro- and nanoelements: “meta-atoms” that mimic electromagnetic reaction of atoms and molecules of natural substances. Meta-atoms are grouped in the form of single or multilayered lattice, and their small size (much less than wavelength of radiation) makes it possible to treat the created lattice as a homogeneous medium for a given wavelength (by analogy with natural crystals); using the concept of “effective medium” for characteristics calculating. New materials can be used in the development of new types of radioelectronic and photonic functional electronics: devices with negative refraction for controlling radiation in gigahertz and visible ranges that allow obtaining of a clear image of elements with dimensions much shorter than the wavelength without diffraction distortion; systems for electromagnetic invisibility, Stealth technology, and much more. The negative refractive index n (Fig. 1.42A) is due to a strong spatial dispersion in metamaterial and to negative values of permittivity and permeability: n ¼ √(εμ) < 0. Because ε(ω) < 0 and μ(ω) < 0, these materials are sometimes called “doubly negative.” (Correspondingly, in conventional materials permeability and permittivity have positive sign; therefore ordinary media sometimes are called “doubly positive.”) The phase velocity of waves in the metamaterials is directed in the opposite direction relatively to group velocity; therefore these materials are also called “backward-wave media.” Metamaterials, interacting with optical frequency radiation, usually are called photonic or optical metamaterials. The main way to obtain metamaterials is based on their “assembly” of huge number of miniature discrete modules, cells, or nanoparticles. These modules (cells and nanoparticles) are sometimes called meta-atoms. It is clear that they are not real atoms, but consist of them, that is, they are made of ordinary matter—mainly metals and dielectrics. Dimensions of meta-atoms greatly exceed atomic dimensions. They form a spatial structure (matrix), for example, an artificial crystal lattice, so that the number of meta-atoms even in a small piece of metamaterial reaches 103–109 (Fig. 1.43). It should be noted here that meta-atoms do not have any chemical bond with each other, unlike atoms of ordinary materials. Therefore the difference in technologies of conventional materials and metamaterials production is understandable. The former are obtained by chemical synthesis from atoms of chemical elements, the latter are obtained as an assemblage of artificial elements by methods of micro- and nanotechnologies. Moreover, it is important that, for incident radiation, the metamaterial imitates a homogeneous medium; for this, the dimensions of meta-atoms and distances between them should be selected to be less than the working wavelength of radiation; the smaller the dimensions, the better the homogeneity condition. Externally, meta-atoms are tiny formations of wires, strips, plates, rods, disks, rings, spirals, balls, films, coatings, and multilayer structures. The millimeter-sized high-ε dielectric resonators and micrometer-sized piezoelectric resonators can serve as dielectric meta-atoms. Moreover, meta-atoms can be in the form of nanoclusters;

1.6 Structures of composites and metamaterials

FIG. 1.43 Practically implemented metastructure for research in microwave frequencies.

finally, they can be a system of holes in flat elements (e.g., they may resemble a fish net). It is important that the configuration and properties of meta-atoms (capacitors, inductances of oscillatory circuits, or miniature resonators) ensure that they perform functions of simplest capacitors, inductances, oscillatory circuits, or miniature (nano-) resonators. Thin layers of metamaterials deposited on a substrate are called metafilms or metacoverings. In the simplest case, the metafilm is a patterned single-layer film made of metal, semiconductor, dielectric, or magnetic material that is deposited on a dielectric or semiconductor substrate. The pattern is determined by the configuration of the abovementioned electroradio elements with unique properties due to resonant interaction with an electromagnetic field. Thus metamaterials are artificial periodic structures with lattice constants much smaller than the wavelength of incident radiation. These are media consisting of resonance elements in which negative propagation of waves takes place. The dimensions of meta-atoms are smaller than the wavelength of radiation interacting with them. They have the ability to simulate homogeneous material, whose properties are absent in natural materials. It is important to note that metamaterials in the optical wavelength range have already been created, and they opened the door to create a new photonic and quantum-optical technology—optical nanoantennes, nanolasers, nonlinear elements, and other devices for generating and controlling light-transmitted systems developed to overcome the diffraction limit. Metamaterials are the basis for such areas of science and technology as nanoplasmonics and nanophotonics. A new class of

61

62

CHAPTER 1 Structure of electronic materials

composite materials has become widespread, in which the scale level of individual component sizes reaches the nanometer range. A nanocomposite is defined as the multicomponent solid material in which one of components in the 1, 2, or 3 dimensions has a size not exceeding 100 nanometers; moreover, nanocomposites are understood as structures consisting of a set of repeating component layers (phases), the distance between which is measured in dozens of nanometers [12]. For example, a method for creating anodes from silicon nanospheres and carbon nanoparticles for lithium batteries has been invented. Anodes made from a siliconcarbon nanocomposite are much more closely adhered to the lithium electrolyte, thereby reducing the charging or discharging time of a device. From nanocomposites, consisting of cellulose base and nanotubes, it is possible to produce conductive paper. If such a paper is placed in the electrolyte, something like a flexible battery will appear. Furthermore, in the electronics industry, nanocomposites are used to produce thermoelectric materials that demonstrate a combination of high electrical conductivity with low thermal conductivity. Graphene occupies a special place in the development of nanocomposite materials. Nanocomposites containing graphene and tin can significantly increase the capacity of lithium-ion batteries and reduce their weight. Recently, it has been found that the addition of graphene to epoxy composites leads to an increase in rigidity and strength of material compared to composites containing carbon nanotubes. Graphene is better combined with epoxy polymer, more effectively penetrating the structure of composite. Nanocomposites based on polymeric matrices and nanotubes are able to change their electrical conductivity due to the displacement of nanotubes relative to each other under the influence of external factors. This property can be used to create microscopic sensors that determine the intensity of mechanical action over extremely short periods of time. Moreover, nanotechnology can be used to produce photonic crystals. Photonic crystals are nanostructured materials in which the periodic change in the refractive index at wavelength scales of visible light creates so-called forbidden bands for photons. These bands influence the propagation of photons of visible light in a material (this effect is similar to how periodic potential in semiconductors affects the determination of the electron flux allowed and the forbidden energy bands). The structure of a photonic crystal can be characterized by a periodic change in the refractive index in 1, 2, or 3 spatial directions. Photonic crystals can be used in the light sources on single crystal, because the pattern of their radiation as well as the direction of beam propagation can be easily controlled.

1.7 SUMMARY 1.

Solids are primarily crystals and polycrystals, as well as ceramics, glasses, glass-ceramics, quasicrystals, amorphous substances, composites, and nanocrystalline structures.

1.7 Summary

2.

Crystals are characterized by a near-perfect well-ordered internal structure. Therefore crystals can be described by 3D spatial periodic structure. A peculiar property of crystals is their translational regulation—elementary cell that consists of a few atoms can be translated supposedly “infinitely” in all directions, creating a regular crystal lattice. 3. Polycrystals consist of a large number of small crystals (crystallites). Macroscopic structure of polycrystals, outwardly, seems disordered, but microscopic components of this structure (crystallites units) are small crystals with perfect microscopic structure and similar properties as a large single crystal. 4. The glass-like and amorphous states of solids are characterized by the absence of long-distant (translational) symmetry. However, these materials are characterized by the order in the immediate surroundings adjacent to each atom. 5. In 2D systems, the strictly ordered structure is possible only in a plane. In such a system, if the planar regularity is repeated, the nanodimensional superstructure (artificially created in semiconductor) can have peculiar electronic properties, characterized by the so-called quantum wells (this case relates to 2D nanostructures). 6. The 1D nanostructures might be linear (wire-like) systems, wherein translated ordering is observed along a single direction. 7. There are, furthermore, systems wherein the dimensions along all three directions are commensurate with the distance between atoms. Such zerodimensional (0D) systems can be “quantum dots,” wherein only 10–103 atoms have an ordered structure. 8. The creation of ordered crystalline (and other) bodies of atoms is accompanied by a decrease in energy. This corresponds to the certain minimum of a system’s energy when atoms become ordered relative to each other, with significant redistribution in electronic density. 9. According to the electronic theory of valence, the interatomic bond occurs due to the redistribution of valence electronic orbitals, and that results in the stable electronic configuration of noble gas (octet) through the formation of ions as well as by the formation of electron pairs between atoms. 10. Any connections of atoms, molecules, or ions are carried out through electrical interaction. At relatively large distances between particles, the electrical forces of attraction dominate whereas, at small distances, the repulsion between particles dramatically increases. The balance between longrange attraction and short-range repulsion determines the basic properties of a certain solid. The bond that occurs between atoms (as a result of spatial restructuring of their valence electrons) and which is caused by these electron interactions is the chemical bond. 11. At the heart of the classification of solids into metals, dielectrics, and semiconductors is the spatial distribution of valence electrons. In molecular crystals, for instance, electrons are completely locked within their molecules. When crystals are formed from atoms of metal, the orbits of

63

64

CHAPTER 1 Structure of electronic materials

12.

13.

14.

15.

16.

17.

valence electrons strongly overlap each other. As a result, valence electrons become distributed within the spaces between atoms and can be described by a general wave function. It is believed that, in metals, an electronic gas is formed. The ionic crystals are chemical compounds that are formed by metallic and nonmetallic elements. The forces of ion attraction are mostly long range: the energy of attraction rather slowly varies with distance. Like molecular crystals, ionic crystals are characterized by such a distribution of electronic charges wherein they are almost completely localized near the ions. The covalent crystals have, in principle, a similar nature of connection as the metals-valence electrons become shared between atoms. The forces of attraction in case of covalent bonds are not so long range as in the case of ionic bonding. The covalent bond (otherwise called a homeopolar bond) is formed by the overlapping (socialization) of pairs of valence electrons. This link is provided with electronic clouds that are called a mutual electron pair. At covalent chemical bond formation, the reduction of total energy occurs due to an exchange interaction that plays an important role in this process. The van der Waals bonds are always present in atomic connections, but they dominate only in the absence of valence bonds; in such cases, these bonds become a principal type of chemical bonding (usually, in molecular crystals). The van der Waals forces of attraction are relatively short range and weak as compared with conventional valence forces. In nonpolar molecules, the forces of attraction arise by mutual deformation of electronic shells. Because this mechanism is investigated through optical polarization dispersion, the forces of attraction of this type are dispersive ones. In polar molecules, the orientation interaction contributes to the energy of bonding. Moreover, there exists an induction interaction between the permanent dipole of one molecule and the induced dipole of another molecule. The hydrogen bonds are realized when two hydrogen atoms in one molecule interact or when a hydrogen atom in one molecule interacts with an electronegative atom such as P, O, N, Cl, or S of another molecule. The cause of the hydrogen bond is the redistribution of electronic density between atoms, induced by the small size of the hydrogen ion (H+; proton). The defects in crystals are formed during their growth (under the influence of thermal, mechanical, and electrical fields), as well as during crystal irradiation by neutrons, electrons, X-rays, and ultraviolet radiation (radiation defects). There are point defects (zero-dimensional), linear (1D) defects, plane defects (2D), and bulk (3D) defects. Parts of atoms or ions of a crystal may be missing locally, thereby violating an ideal crystal lattice scheme: these defective places are the vacancies. Furthermore, foreign (impurity) atoms or ions can exist in crystals, replacing basic particles that form a crystal or take root between them. Their own atoms (or ions) can serve as point defects in crystals, if they shift from normal positions (interstitial atoms or ions).

1.7 Summary

18.

19.

20.

21.

22.

In the process of crystal growth or during its plastic deformation and in many other cases, dislocations can appear. Moreover, dislocations can arise in a crystal during its doping. The distribution and behavior of dislocations under external influences determine many important mechanical properties of a crystal, including strength, ductility, and so on. The mobility of dislocation determines the plasticity of crystals; at the location of the greatest internal stress, clusters of dislocation can occur that can cause the destruction of the crystal. The problem of plastic flow (i.e., irreversible deformation) in metallic crystals is solved by the association and movement of dislocations. These dislocations impede the process of magnetization and electrical polarization due to their interaction with domain boundaries. Elastic deformations of crystal structure arise in the vicinity of defects that lead, in turn, to the appearance of internal mechanical stresses. For example, point defects interact with dislocations that result in the increase or decrease of crystal strength. Defects in crystals affect the absorption spectra of luminescence, light scattering in a crystal and can change electrical conductivity, thermal conductivity, ferroelectric and ferromagnetic properties, etc. The vacancies in crystal lattice usually are Schottky defects. The formation of vacancies can be explained by some atoms moving outside from the crystal surface and they being replaced by other atoms from a volume. For most crystals, the energy of vacancy formation is approximately 1 eV. Lattice defects that usually are called the Frenkel defects arise by mechanisms that generate interstitial atoms or ions in a crystal. The polarons are charge carriers bound in the lattice of an ionic crystal (most often, they are bound electrons). The polaron is not a “static” defect because it is much more mobile than vacancies or interstitial ions. The excitons can be interpreted as the mobile point defects in a crystal. In the case of excitons, atoms or ions of crystal do not change their location, but they become significantly different from their neighbors by excited electronic states. The movement of an exciton in a crystal is not connected with the change of atom or ion positions, and therefore excitons (as polarons) have much greater mobility than replacements of vacancies, interstitial atoms, and impurities. The dislocations are crystallographic defects or irregularities within crystal structure. The presence of dislocations strongly influences many properties of materials. The edge dislocation is a land of “excessive” atomic planes that splits the crystal. It corresponds to the row of ordinary atoms along the edge of an additional part-plane of atoms within the crystal. In other words, edge dislocation is such a defect wherein an extra half-plane of atoms can move through the crystal, distorting the nearby planes of atoms. The screw dislocation is a result of changes of one area of crystal with regard to another. It corresponds to the spiral axis of structural distortion, connected to normal parallel planes. It comprises the structure wherein a helical path is traced around a linear defect (dislocation line).

65

66

CHAPTER 1 Structure of electronic materials

23.

24.

25.

26.

27.

28.

The solid solutions are widely used in electronic components technology. The presence of two components is possible in crystals or polycrystals (in metals, these are alloys). The solid-state solution is a mixture that remains in a single homogeneous phase. The interstitial type of solid solutions is a result of the fact that atoms of the element, which dissolves, are placed in empty spaces of the solvent lattice. The substitutional solid solutions are formed by a partial substitution of solvent atoms. This process can occur without incurring significant stresses in structure only when the size of atoms does not differ greatly among themselves. A structure is polytypic when it is composed of similar structural elements but with a different sequence of their location. Polytypic lattice parameters in a plane layer are unchanged but, in the direction perpendicular to layers, lattice parameters are different, although they are always multiples of the distance between adjacent layers. Polytypism is a special case of polymorphism: 2D translations within layers are essentially preserved. Isomorphism and polymorphism. The property of chemically closed atoms, ions, or other structural elements to replace each other in the crystal lattice and form continuously variable composition is isomorphism. The ability of certain substances to exist in multiple crystalline phases, differing in symmetry of structure and in physical properties, is polymorphism. The change in environmental conditions may cause polymorphous transformation. During these transformations (that usually are phase transitions of first order), heat absorption and internal energy jumps are observed as well as changes in other physical properties of matter. Furthermore, there are such polymorphic modifications that differ by very little changes in physical properties. Polymorphic transitions between states are phase transitions of second order and usually are described as “order-disorder” type of transitions. The symmetry of crystal structures determines their physical properties. Therefore many properties of solids may be described by the peculiarities of crystal symmetry. The relationship between the geometry of external shape and internal building of crystals, as well as their physical properties, are specified by physical crystallography. The physics of crystals formulates some principles that establish a connection between the symmetry of a crystal and physical phenomena; central to these are Neumann principle and Curie principle. The mechanism of how the physical properties of crystals are conditioned with their symmetry was formulated by Neumann: the symmetry of physical properties of a crystal is not lower than the symmetry of its structure. This means that the structure of a crystal contains all elements of the symmetry of its properties (but also may have other symmetry elements). Therefore, information about crystal symmetry enables prediction of the possible physical effects in a crystal. In accordance with the Curie principle the crystal, being under external influence, has only those symmetry elements that are common to the crystal in

1.7 Summary

29.

30.

31.

32.

33.

34.

35.

the absence of influence and impact (in case of lack of crystal)—that is, in the system “crystal-influence,” only common elements of symmetry remain. As the element of symmetry, an imaginary object can be used that supports the realization of the operation of symmetry. To such elements of symmetry belong the planes, axis, and center of symmetry (center of the inversion). The combination of a point (or part of figure) with another point (or part of figure) is called an operation of symmetry. Both parts of figures that are combined are symmetric. Operations of point symmetry are left in its place, at least, on one point of the final figure. This is the point of intersection of all elements of symmetry. Rotation and mirror rotation as well as inverted rotations and reflections in the plane of symmetry are selected as symmetric operations. There are elements of symmetry of the first and second kinds. The former include the symmetry plane, rotary axis, and center of inversion (symmetry); the second include some complex elements of symmetry: inversions and mirror-rotary axes. To analyze symmetry, screw rotations and/or glide reflections are also used. These are rotations or reflections, together with partial translation. The Bravais lattices may be considered the outcome of translational symmetry operations. Combinations of operations with additional symmetry operations produce 230 crystallographic space groups. The plane of symmetry is a plane of mirror reflection; this is an operation of a similar point combination. To refer to a specific class of symmetry elements, the plane of symmetry can be denoted by P. In the international system, a mirrored plane is denoted by the letter m, it bisects all segments that connect symmetric points that are perpendicular to it (part of the figure). Rotational symmetry is symmetry with respect to some or all rotations in the Euclidean space. The rotary axis of symmetry of the nth order is denoted as Ln, that rotates around a certain angle α ¼ 360°/n. Moreover, the rotary axes are marked by symbols 1, 2, 3, 4, 5, 6, 7, …, ∞, where the numbers indicate the order of axis. The n-fold rotational symmetry operation rotates the object by 360°/n. Only n ¼ 1, 2, 3, 4, and 6 are permitted in the periodic lattice. The inversion axis is a combination of rotation and the center of symmetry operations. The center of symmetry (inversion center, denoted as C) is a special point inside a figure or unit cell; it is characterized by the fact that any line drawn through the center of symmetry falls into the same point of figures on both sides of the center at equal distances. The class of symmetry is a set of symmetry elements of the crystal (or any object) that describes its possible symmetric transformations. A unit cell can be selected in any crystal and, on its basis, all crystal lattices can be built using translations. These translations are the displacement of a unit cell within a crystal. The full set of symmetry elements of any material is known as the group of symmetry. Crystals and textures that have a center of symmetry cannot show piezoelectric properties. In the absence of external influences, only noncentrosymmetric

67

68

CHAPTER 1 Structure of electronic materials

36.

37.

38.

39.

40.

structures are capable of being piezoelectrics. Among them, only the crystal with a polar axis might be pyroelectric. The quasicrystals exhibit a special, new type of symmetry, different from the usual crystals. They have symmetry elements that previously were considered impossible in crystals: classic crystallography does not allow symmetry axes of fifth, seventh, and higher orders. With these axes, the elementary cell cannot ensure complete filling even on the plane (and, moreover, in volume). However, quasicrystals exist, and they can have axes of symmetry of fifth, eighth, or higher orders. Nanomaterials, as a rule, are small particles (clusters) of materials consisting of 10–1000 atoms. Their properties depend on the number of atoms in the cluster and on the relative position of atoms, as well as on the shape and symmetry of clusters. The composites consist of different materials united in a single whole, and have important applications in electronic devices. They are used in various active and passive components (e.g., piezoelectric with polymer). The physical and technical properties of composites that ensure their applications, with advantages over crystals, ceramics, and polymeric materials, can be described by three effects: the sum effect, the combinative effect, and the effect of the product. Electromagnetic metamaterials are artificially structured in a special way to be mediums that have electrical and magnetic properties, which are significantly different from the original structural materials. For example, a metamaterial can have a negative refractive index, which is never observed in natural materials. The internal structure of metamaterials plays an important role in the formation of their characteristics and parameters. Nanocomposites are solid formations consisting of a basic matrix and nanosized components that differ in their structural parameters and chemical properties. Mechanical, electrical, thermal, optical, and other characteristics of nanocomposites differ significantly from the properties of ordinary composite materials made of the same basic substances or elements.

REFERENCES [1] L.H. Van Vlack, Materials Science for Engineers, Addison-Wesley Publishing Co., Reading, MA, 1975. [2] A. Holden, Bonds Between Atoms, Oxford University Press, New York, 1970. [3] C. Kittel, Introduction to Solid State Physics, fifth ed., John Willey, New York, 1976. [4] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt and Winston, New York, 1976. [5] Y.M. Poplavko, Polar Crystals: Physical Nature and New Effects, Lambert Academic Publishing, Saarbrucken, 2014. [6] M.A. Shaskolskaya, Crystallography, Vishaya Shkola, Moscow, 1976. [7] I.S. Jeludev, Symmetry and Its Application, Energoatomizdat, Moscow, 1983. [8] I.V. Zolotuhin, Y.E. Kalinin, O.V. Stogney, New Directions of Physical Materials Science, Voronej University Ed., Voronej, 2000.

References

[9] I.M. Bolesta, Solid State Physics, Lvov University Ed, Ukraine, 2003. [10] A.F. Kravchenko, V.N. Ovsyuk, Electronic Processes in Low-Dimensional Solid Systems, Novosibirsk University Ed, Russia, 2000. [11] R. Waser (Ed.), Nanoelectronics and Information Technology: Advanced Electronic Materials and Novel Devices, Wiley-VCH, Weinheim, 2005. [12] H.S. Nalva (Ed.), Nanostructured Materials and Nanotechnology, Academic Press, New York, 2002. [13] R.E. Newnham, D.P. Skinner, L.E. Cross, Connectivity and piezoelectric-pyroelectric composites, Mater. Res. Bull. 13 (1978) 525. [14] K. Uchino, Ferroelectric Devices, Marcel Dekker, New York, 2000. [15] G.V. Eleftheriades, in: G. Balmain (Ed.), Negative-Refraction Metamaterials, Wiley, Hoboken, NJ, 2005.

69